California Math Triumphs VOL 6B Measurement (CALIFORNIA MATH TRIUMPHS VOL 6B)

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Authors Basich Whitney • Brown • Dawson • Gonsalves • Silbey • Vielhaber

Photo Credits Cover Jupiter Images; iv (tl bl br) File Photo, (tc tr) The McGraw-Hill Companies, (cl c) Doug Martin, (cr) Aaron Haupt; v (1 2 3 4 6 7 8 9 11 12) The McGraw-Hill Companies, (5 10 13 14) File Photo; viii CORBIS; viii Mitchell Funk/Getty Images; ix S. Alden/PhotoLink/Getty Images; xi Peter Barritt/Alamy; 2–3 Mike Brinson/Getty Images; 9 (Frame) Getty Images; 9 (inset) Ryan McVay/Getty Images; 15 Comstock/CORBIS; 16 CORBIS; 25 Jerry Irwin/Photo Researchers, Inc.; 34 42 CORBIS; 52 Getty Images; 53 CORBIS; 061 Ingram Publishing/ SuperStock; 068 (b) CORBIS, (t) Rudi Von Briel/PhotoEdit; 70 71 CORBIS; 77 Getty Images; 78 CORBIS; 79 Comstock/Imagestate; 80 CORBIS; 95 Punchstock

Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without prior permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH 43240-4027 ISBN: 978-0-07-878214 MHID: 0-07-878214-7 Printed in the United States of America. 1 2 3 4 5 6 7 8 9 10 055/027 16 15 14 13 12 11 10 09 08 07

California Math Triumphs Volume 6B

California Math Triumphs Volume 1 Place Value and Basic Number Skills 1A Chapter 1 Counting 1A Chapter 2 Place Value 1A Chapter 3 Addition and Subtraction 1B Chapter 4 Multiplication 1B Chapter 5 Division 1B Chapter 6 Integers Volume 2 Fractions and Decimals 2A Chapter 1 Parts of a Whole 2A Chapter 2 Equivalence of Fractions 2B Chapter 3 Operations with Fractions 2B Chapter 4 Positive and Negative Fractions and Decimals

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Volume 3 Ratios, Rates, and Percents 3A Chapter 1 Ratios and Rates 3A Chapter 2 Percents, Fractions, and Decimals 3B Chapter 3 Using Percents 3B Chapter 4 Rates and Proportional Reasoning Volume 4 The Core Processes of Mathematics 4A Chapter 1 Operations and Equality 4A Chapter 2 Math Fundamentals 4B Chapter 3 Math Expressions 4B Chapter 4 Linear Equations 4B Chapter 5 Inequalities Volume 5 Functions and Equations 5A Chapter 1 Patterns and Relationships 5A Chapter 2 Graphing 5B Chapter 3 Proportional Relationships 5B Chapter 4 The Relationship Between Graphs and Functions Volume 6 Measurement 6A Chapter 1 How Measurements Are Made 6A Chapter 2 Length and Area in the Real World 6B Chapter 3 Exact Measures in Geometry 6B Chapter 4 Angles and Circles iii

Authors and Consultants AUTHORS

Frances Basich Whitney

Kathleen M. Brown

Dixie Dawson

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Math Curriculum Leader Long Beach Unified Long Beach, California

Philip Gonsalves

Robyn Silbey

Kathy Vielhaber

Mathematics Coordinator Alameda County Office of Education Hayward, California

Math Specialist Montgomery County Public Schools Gaithersburg, Maryland

Mathematics Consultant St. Louis, Missouri

Viken Hovsepian Professor of Mathematics Rio Hondo College Whittier, California

Dinah Zike Educational Consultant, Dinah-Might Activities, Inc. San Antonio, Texas

CONSULTANTS Assessment Donna M. Kopenski, Ed.D. Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

iv

Instructional Planning and Support

ELL Support and Vocabulary

Beatrice Luchin

ReLeah Cossett Lent

Mathematics Consultant League City, Texas

Author/Educational Consultant Alford, Florida


CONTRIBUTING AUTHORS

California Advisory Board CALIFORNIA ADVISORY BOARD


Glencoe wishes to thank the following professionals for their invaluable feedback during the development of the program. They reviewed the table of contents, the prototype of the Student Study Guide, the prototype of the Teacher Wraparound Edition, and the professional development plan.

Linda Anderson

Cheryl L. Avalos

Bonnie Awes

Kathleen M. Brown

4th/5th Grade Teacher Oliveira Elementary School, Fremont, California

Mathematics Consultant Retired Teacher Hacienda Heights, California

Teacher, 6th Grade Math Monroe Clark Middle School San Diego, California

Math Curriculum Staff Developer Washington Middle School Long Beach, California

Carol Cronk

Audrey M. Day

Jill Fetters

Grant A. Fraser, Ph.D.

Mathematics Program Specialist San Bernardino City Unified School District San Bernardino, California

Classroom Teacher Rosa Parks Elementary School San Diego, California

Math Teacher Tevis Jr. High School Bakersfield, California

Professor of Mathematics California State University, Los Angeles Los Angeles, California

Eric Kimmel

Donna M. Kopenski, Ed.D.

Michael A. Pease

Chuck Podhorsky, Ph.D.

Mathematics Department Chair Frontier High School Bakersfield, California

Math Coordinator K–5 City Heights Educational Collaborative San Diego, California

Instructional Math Coach Aspire Public Schools Oakland, California

Math Director City Heights Educational Collaborative San Diego, California

Arthur K. Wayman, Ph.D.

Frances Basich Whitney

Mario Borrayo

Melissa Bray

Professor Emeritus California State University, Long Beach Long Beach, California

Project Director, Mathematics K–12 Santa Cruz County Office of Education Capitola, CA

Teacher Rosa Parks Elementary San Diego, California

K–8 Math Resource Teacher Modesto City Schools Modesto, California

v

California Reviewers CALIFORNIA REVIEWERS Each California Reviewer reviewed at least two chapters of the Student Study Guides, providing feedback and suggestions for improving the effectiveness of the mathematics instruction. Melody McGuire

Math Teacher California College Preparatory Academy Oakland, California

6th and 7th Grade Math Teacher McKinleyville Middle School McKinleyville, California

Eppie Leamy Chung

Monica S. Patterson

Teacher Modesto City Schools Modesto, California

Educator Aspire Public Schools Modesto, California

Judy Descoteaux

Rechelle Pearlman

Mathematics Teacher Thornton Junior High School Fremont, California

4th Grade Teacher Wanda Hirsch Elementary School Tracy, California

Paul J. Fogarty

Armida Picon

Mathematics Lead Aspire Public Schools Modesto, California

5th Grade Teacher Mineral King School Visalia, California

Lisa Majarian

Anthony J. Solina

Classroom Teacher Cottonwood Creek Elementary Visalia, California

Lead Educator Aspire Public Schools Stockton, California

vi


Bobbi Anne Barnowsky

Volume 6A

Measurement

Chapter

How Measurements Are Made

1

1-1 Unit Conversions: Metric Length ...................................4. 3AF1.4, 3MG1.4, 6AF2.1

1-2 Unit Conversions: Customary Length .........................11 3AF1.4, 3MG1.4, 6AF2.1

Progress Check 1 .............................................................18 1-3 Unit Conversions: Metric Capacity and Mass ............19 3AF1.4, 3MG1.4, 6AF2.1, 7MG1.1

1-4 Unit Conversions: Customary Capacity and Weight… ...................................................................25 3AF1.4, 3MG1.4, 6AF2.1

Progress Check 2 .............................................................32 1-5 Time and Temperature ...................................................33 3AF1.4, 3MG1.4, 6AF2.1, 7MG1.1

1-6 Analyze Units of Measure .............................................39


6AF2.1, 7MG1.1, 7MG1.3

Progress Check 3 .............................................................46 Assessment Study Guide .....................................................................47

Chapters 1 and 2 are contained in Volume 6A. Chapters 3 and 4 are contained in Volume 6B.

Standards Addressed in This Chapter 3AF1.4 Express simple unit conversions in symbolic form (e.g., ___ inches = ___ feet × 12). 3MG1.4 Carry out simple unit conversions within a system of measurement (e.g., centimeters and meters, hours and minutes). 6AF2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). 7MG1.1 Compare weights, capacities, geometric measures, times, and temperatures within and between measurement systems. (e.g., miles per hour and feet per second, cubic inches to cubic centimeters) 7MG1.3 Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; and use dimensional analysis to check the reasonableness of the answer.

Chapter Test .....................................................................50 Standards Practice ...................................................52

Lake Tahoe

vii

Contents Chapter

Length and Area in the Real World

2

Standards Addressed in This Chapter 2-1 Length ..............................................................................56 2MG1.3, 4MG2.2, 4MG2.3

2-2 Perimeter ..........................................................................63 3MG1.3

Progress Check 1 .............................................................70 2-3 Introduction to Area .......................................................71 3MG1.2

2-4 Introduction to Volume ................................................. 77 3MG1.2

Progress Check 2 .............................................................83 Assessment

2MG1.3 Measure the length of an object to the nearest inch and/or centimeter. 3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 3MG1.3 Find the perimeter of a polygon with integer sides. 4MG2.2 Understand that the length of a horizontal line segment equals the difference of the x-coordinates. 4MG2.3 Understand that the length of a vertical line segment equals the difference of the y-coordinates.

Study Guide .....................................................................84 Chapter Test .....................................................................88 Standards Practice ...................................................90


viii

Alamo Square, San Francisco

Contents Chapter

Exact Measures in Geometry

3

3-1 Area of a Rectangle ..........................................................4 3MG1.2, 4MG1.1

3-2 Area of a Parallelogram..................................................11 4MG1.1, 5MG1.1

Progress Check 1 .............................................................18 3-3 Area of a Triangle ............................................................19 3MG1.2, 5MG1.1

3-4 Surface Area of Rectangular Solids ............................. 27 3MG1.2, 4MG1.1, 5MG1.2

Progress Check 2 .............................................................36 3-5 Volume of Rectangular Solids .......................................37 3MG1.2, 5MG1.3

Assessment


Study Guide .....................................................................43 Chapter Test .....................................................................48 Standards Practice ...................................................50 Santa Cruz

Chapters 1 and 2 are contained in Volume 6A. Chapters 3 and 4 are contained in Volume 6B.

Standards Addressed in This Chapter 3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in.2), square yard (yd.2), or square mile (mi.2). 5MG1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle (i.e., two of the same triangles make a parallelogram with twice the area; a parallelogram is compared with a rectangle of the same area by pasting and cutting a right triangle on the parallelogram). 5MG1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to complete the surface area for these objects. 5MG1.3 Understand the concept of volume and use the appropriate units in common measuring systems (i.e., cubic centimeter [cm3], cubic meter [m3], cubic inch [in.3], cubic yard [yd.3]) to compute the volume of rectangular solids.

ix

Contents Chapter

Angles and Circles

4

Standards Addressed in This Chapter 4-1 Lines 5MG2.1 ....................................................................54 4-2 Angles 5MG2.1 .................................................................63 Progress Check 1.............................................................72 4-3 Triangles and Quadrilaterals 5MG2.1 ...........................73 4-4 Add Angles 5MG2.1, 5MG2.2, 6MG2.2 ............................. 81 Progress Check 2.............................................................90 4-5 Congruent Figures 7MG3.4 ............................................91 4-6 Pythagorean Theorem 5MG2.1, 7MG3.3 ........................ 99 Progress Check 3...........................................................108 4-7 Circles 6MG1.2 ...............................................................109 4-8 Volume of Triangular Prisms and Cylinders ........... 117 6MG1.3

Progress Check 4...........................................................127

Study Guide ..................................................................128 Chapter Test ..................................................................134 Standards Practice.................................................136 Mono Lake Tufa State Reserve

5MG2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems. 6MG1.2 Know common estimates 22 of π (3.14, ___) and use these values to 7 estimate and calculate the circumference and the area of circles; compare with actual measurements. 6MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid. 6MG2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle. 7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. 7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationship between the sides and angles of the two figures.

x


Assessment

5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straight edge, ruler, compass, protractor, drawing software).

R E G N E V A SC HUNT Let’s Get Started Use the Scavenger Hunt below to learn where things are located in each chapter. 1 What is the title of Chapter 4? 2

Angles and Circles

What is the Key Concept of Lesson 3-3?

Area of a Triangle 3

What is the definition of surface area on page 27?

the area of the surface of a three-dimensional figure 4

What are the vocabulary words for Lesson 4-4?


complementary angles, supplementary angles 5

How many Examples are presented in the Chapter 4 Study Guide? 9

6

What are the California Standards covered in Lesson 4-3?

5MG2.1 7

According to Lesson 3-1, what is the formula for the area of a rectangle? A = × w

8

What do you think is the purpose of the Progress Check on page 36? It allows you to practice problems from the last

two lessons. 9

On what pages will you find the Study Guide for Chapter 4?

pages 126–131 10

In Chapter 3, find the logo and Internet address that tells you where you can take the Online Readiness Quiz. It is

found on page 3. The URL is ca.mathtriumphs.com. 1

Chapter

3

Exact Measures in Geometry Why is area important? What if you compare your statistics in volleyball with your cousin’s, but the two of you play on different-sized courts? It would be an unfair comparison. Most school teams play on volleyball courts that are of regulation size.

Copyright © by The McGraw-Hill Companies, Inc.

2

Chapter 3 Exact Measures in Geometry

Mike Brinson/Getty Images

STEP

STEP

1 Quiz

Are you ready for Chapter 3? Take the Online Readiness www.mathtriumphs.com to to find find out. out. Quiz at ca.mathtriumphs.com

2 Preview

Get ready for Chapter 3. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know how to measure the lengths of items.

5IFMFOHUIPGUIJT SFDUBOHMFJTJO

JO JO

You know that you can make a parallelogram into a rectangle. Step 1: Step 2: Copyright © by The McGraw-Hill Companies, Inc.

To find the area of a rectangle, you multiply the length by the width. Area = length × width Area = 2 inches × 1 inch Area = 2 square inches

JO

5IFXJEUI PGUIJT SFDUBOHMF JO JTJO

Lesson 3-1

Lesson 3-2 To find the area of a parallelogram, you multiply the base by the height. Area = base × height Area = 3 inches × 2 inches Area = 6 square inches JO

Step 3: JO

You know you can separate a rectangle into two triangles.

Lesson 3-3 A triangle is half of a rectangle. So the area of a triangle is one-half the area of a rectangle. "SFBCBTFIFJHIU

"SFBCBTFIFJHIU

3

Lesson

3-1 Area of a Rectangle KEY Concept Find the area of a rectangle using the formula below. is the length of the rectangle. A is the area of the rectangle.

A=×w

AK

ŰVU

w is the width of the rectangle.

AKVAKAK

3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2).

VOCABULARY area the number of square units needed to cover a region or a plane figure (Lesson 2-3, p. 71)

square unit a unit for measuring area

AK

(Lesson 2-3, p. 71)

rectangle a quadrilateral with four right angles; opposite sides are parallel and equal in length

The area of the rectangle is 12 square centimeters.

The units of area are square units.

JO JO

JO JO

square a rectangle with congruent sides

What is the area of the rectangle?

ZE

1. The length of the rectangle is 5 inches, and the width is 3 inches.

GL GL

2. Substitute these values into the formula. Multiply. A=×w A = 5 in. × 3 in. A = 15 in2 The area of the rectangle is 15 square inches.

4


ZE

ZE ZE


Example 1

YOUR TURN! What is the area of the rectangle? 1. The length of the rectangle is width is 2 yards.

7

yards, and the

WB WB

2. Substitute these values into the formula. Multiply. A=×w A=

7

yd ×

A=

14

yd2

2

yd

The area of the rectangle is

14

square yards.

Example 2

YOUR TURN! What is the area of the square?

What is the area of the square?

1. The length of the square is 5 kilometers, and the width is 5 kilometers.

1. The length of the square is 4 feet, and the width is 4 feet.


2. Substitute these values into the formula. Multiply.

DR

IK

IK


DR

A=×w A = 4 ft × 4 ft A = 16 ft2

A=×w A = 5 km × A = 25 km2

The area of the square is 16 square feet.

5

The area of the square is kilometers.

km

25

square

Who is Correct? What is the area of the rectangle at the right?

Bena 2 A = 4 + 8 = 12 mi

Ryan 2

A = 4 × 8 = 32 mi

Arturo

KG

2 A = 8 × 8 = 64 mi

KG

Circle correct answer(s). Cross out incorrect answer(s). Lesson 3-1 Area of a Rectangle

5

Guided Practice Draw a rectangle for each given area. 1

18 cm2

Sample answer:

2

4 cm2

Sample answer: AK

AK AK

AK

Step by Step Practice 3

What is the area of the rectangle?

GU

6

Step 1 The length of the rectangle is 5 feet.

feet, and the width

GU

Step 2 Substitute these values into the formula. Multiply. A=l×w

6 30

A= A=

ft ×

5

ft

2

ft


30

square feet.

4

The length of the rectangle is 6 inches.

8

JO

inches. The width is JO

A=l×w A= A= 5

in. ×

A=

9 27

6

in.

2

in

The length is A=

6

8 48

9

m×

meters. The width is

3

m

m2


3

meters.

N N


Find the area of each rectangle.

6

ZE

36 yd2

A=

7

2 A = 100 cm

DN

ZE DN

8

40 mi2

A=

9

A=

36 km2

NJ

LN LN NJ

Step by Step Problem-Solving Practice

Problem-Solving Strategies ✓ Use a formula.

Solve. 10

Understand


Look for a pattern. Guess and check. Act it out. Solve a simpler problem.

BASKETBALL A high-school basketball court is 94 feet long and 50 feet wide. What is the area of the court? Read the problem. Write what you know. The length of the basketball court is 94 and the width of the court is 50 feet. Plan

feet,

Pick a strategy. One strategy is to use a formula. Substitute values for length and width into the area formula.

Solve

Use the formula. A=×w A=

94

ft ×

50

ft

A = 4,700 ft2 The area of the high-school basketball court is 4,700 square feet. Check

Use a calculator to check your answer. GO ON Lesson 3-1 Area of a Rectangle

7

11

CONSTRUCTION A construction crew is pouring cement for sidewalk slabs. Each slab is a square that has sides that measure 2 70 centimeters. What is the area of each slab? 4,900 cm Check off each step.

12

✔

Understand

✔

Plan

✔

Solve

✔

Check

ART Mrs. Brady asked her class to use an entire sheet of paper to finger paint. Each sheet of paper measured 8.5 inches by 11 inches. What was the area of the finger painting?

93.5 in2 13

Can two rectangles have the same area but different lengths and widths? Explain.

Sample answer: Yes; A rectangle with side lengths of 1 inch and 6 inches has an area of 6 in2. A rectangle with side lengths of 2 inches and 3 inches also has an area of 6 in2.

Draw a rectangle that has the given area. 14

8

6 square units Sample answer:


15

24 square units Sample answer:


Skills, Concepts, and Problem Solving

Find the area of each rectangle. 16

16 m2

A=

17

A=

N

28 mi2 NJ

N NJ

18

18 yd2

A=

19

A=

36 in2 JO

ZE ZE

JO

Solve. 20

28 cm

PHOTOS At the portrait studio, Ines ordered the picture of her family shown. What was the area of Ines’s family portrait?

1,008 cm2 21

DOORS The screen door at Ethan’s house is 32 inches wide and 85 inches tall. What is the area of Ethan’s screen door?

36 cm


2,720 in2 Vocabulary Check Write the vocabulary word that completes each sentence.

Area

22

is the number of square units needed to cover the inside of a region or a plane figure.

23

A(n) sides.

square

is a rectangle with four congruent

24

rectangle A(n) has opposite sides that are equal and parallel. It is a quadrilateral with four right angles.

25

Writing in Math Explain how to find the area of a rectangle.

Sample answer: Identify the length and width of the rectangle. Substitute values of the length and width into the formula for the area of a rectangle. Multiply to find the area of the rectangle. Express the answer in square units.

GO ON

Lesson 3-1 Area of a Rectangle (frame) Getty Images; (inset) Ryan McVay/Getty Images

9

Spiral Review 26

PACKAGES A package is 8 feet long, 5 feet wide, and 4 feet tall. What is the volume of the package?

160 cubic feet 27

(Lesson 2-4, p. 77)

FOOD A cereal box is 8 inches by 10 inches by 3 inches. What is maximum volume of cereal the box can hold?

240 cubic inches 28

GARDENING A flower bed is 3 meters long, 1 meter wide, and 1 meter deep. How many cubic feet of dirt fills the flower bed? __ 2

1.5 cubic meters

Find the perimeter of each polygon. 29

(Lesson 2-2, p. 63)

The perimeter of the square is 28 feet.

30

GU

The perimeter of the square is 8 meters. N

N

GU

The perimeter of the rectangle is 16 inches.

32

The perimeter of the rectangle is 16 inches. DN

JO

DN

JO

Convert.

(Lesson 1-4, p. 25)

33

5,000 lb = 2.5 T

34

52 fl oz = 6.5 c

35

10 c =

5

pt

36

20 qt =

5

gal

37

3 pt =

6

c

38

48 oz =

3

lb

10



31

Lesson

3-2 Area of a Parallelogram

4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2). 5MG1.1 Derive and use the formula for area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle.

KEY Concept

parallelogram

VOCABULARY Cut a triangle from the parallelogram along the dashed line. Move the triangle and place it on the other side, next to the right edge of the parallelogram. In the parallelogram, b represents the base, and h represents the height. C

area the number of square units needed to cover a region or a plane figure (Lesson 2-3, p. 71)

C

I

square unit a unit for measuring area

I

(Lesson 2-3, p. 71)

Notice that the new shape formed from the parallelogram is a rectangle. So, the formula for the area of a parallelogram is similar to the formula for the area of a rectangle: The formula A = b × h is the area of a parallelogram. "ŰX

(Lesson 3-1, p. 4)

5IJTJTMJLFUIFBSFBPGB SFDUBOHMF FYDFQUUIFMFOHUI JTCBOEUIFXJEUIJTI

parallelogram a quadrilateral in which each pair of opposite sides is parallel and equal in length


"CI b is the length of the base.

rectangle a quadrilateral with four right angles; opposite sides are parallel and equal in length

h is the height.

Example 1 What is the area of the parallelogram? 1. The base of the parallelogram is 8 inches, and the height is 7 inches.

GL

2. Substitute these values into the formula. Multiply. A=b×h A = 8 in. × 7 in. A = 56 in2 The area of the parallelogram is 56 square inches.

GL

GO ON

Lesson 3-2 Area of a Parallelogram

11

YOUR TURN! What is the area of the parallelogram?

4

1. The base of the parallelogram is height is 9 feet.

feet, and the

DR

2. Substitute these values into the formula. Multiply. A=b×h A = 4 ft × A = 36 ft2

9

DR

ft

The area of the parallelogram is

36

Example 2

square feet.

YOUR TURN!

What is the area of the parallelogram? 1. The base of the DN parallelogram is 9 centimeters and the DN height is 9 centimeters. 2. Substitute these values into the formula. Multiply.

What is the area of the parallelogram? 1. The base of the parallelogram is 7 yards, and the height is 7 yards.

ZE

ZE


The area of the parallelogram is 81 square centimeters.

7

yd

The area of the parallelogram is square yards.

Who is Correct? What is the area of the parallelogram?

Jen 2 A = 6 × 7 = 42 m

Emilio

Jade

2 A = 6 × 7 = 48 m

2 A = 6 + 7 = 13 m

Circle correct answer(s). Cross out incorrect answer(s).

12


N

N

49


A=b×h A = 7 yd × A = 49 yd2

A=b×h A = 9 cm × 9 cm A = 81 cm2

Guided Practice Draw a parallelogram that has the given area. 1

30 cm2

Sample answer:

2

200 mm2

Sample answer:

KK AK KK AK

Step by Step Practice Find the area of the parallelogram. 3


GL

GL

Step 1 The base of the parallelogram is height is 9 inches.

7

inches, and the

Step 2 Substitute these values into the formula. Multiply. A=b×h A = 7 in. × A = 63 in2

9

in.


63

square inches.

GO ON Lesson 3-2 Area of a Parallelogram

13

Find the area of each parallelogram. 4

The base of the parallelogram is is 9 inches. A=b×h A = 5 in. × A = 45 in2

9

5

inches, and the height

in.


GL

45

square inches. GL

5

LN LN

The base is

4

A = 4 km × 2 A = 8 km

kilometers, and the height is

2

2

kilometers.

km

6

7 NJ DN NJ

72 mi2

DN

A=

80 cm2

8

9 N N

GU

A=

GU

A=

14

100 ft2


9 m2


A=


Problem-Solving Strategies Draw a diagram.

Solve. 10

✓ Use a formula.

HOBBIES Tony bought a sail for his boat. The sail is in the shape of a parallelogram and is 21 feet wide at the base and 42 feet tall. What is the area of the new sail? Understand

Guess and check. Solve a simpler problem. Work backward.

Read the problem. Write what you know. The base of the sail is 21 feet and the height of the sail is 42 feet. Pick a strategy. One strategy is to use a formula.

Plan

Substitute values for base and height into the area formula. Use the formula.

Solve

A=b×h A = 21 × A = 882

HOBBIES A sailboat can have sails shaped like parallelograms.

42

The area of the sail is 882 ft2. Use a calculator to check your multiplication.

Check


11

ART Part of the sculpture that sits in the middle of Jacob Park is shaped like a parallelogram. The front of this piece is 13 feet tall and has a base of 8 feet. What is the area of the front of the 2 sculpture? 104 ft Check off each step.

12

✔

Understand

✔

Plan

✔

Solve

✔

Check

ART Camila’s class is making cardboard ornaments. Each ornament is shaped like a parallelogram with a height of 16 centimeters and a base of 9 centimeters. What is the area of each ornament?

144 cm2

GO ON Lesson 3-2 Area of a Parallelogram

Corbis

15

13

Compare the area of a parallelogram with a base of 10 feet and a height of 5 feet to the area of a rectangle with a length of 25 feet and a width of 2 feet. Explain.

See TWE margin.

Skills, Concepts, and Problem Solving Draw a parallelogram that has the area given. 14

Sample answer:

220 mm2

15

Sample answer:

5 in2

KK

GL

GL

KK


17 N

GU GU

N

44 m2

A=

18

45 ft2

19 DN ZE DN ZE 2

A = 128 cm

A=

96 yd2

Solve. 20

FARMING Mrs. Rockwell’s cornfield is in the shape of a parallelogram. Refer to the photo caption at the right. What is the area of the cornfield?

4,108 m2

16 Corbis


FARMING The height of Mrs. Rockwell’s cornfield is 79 meters and the base is 52 meters.


A=

21

PARTIES Toby made a sandwich. She cut the bread in the shape of a parallelogram. Each piece of bread is 45 millimeters at the base and 65 millimeters tall. What is the area of each piece of bread?

2,925 mm2 Vocabulary Check Write the vocabulary word that completes each sentence. 22 23

24

Square unit

is a unit for measuring area.

parallelogram A(n) is a quadrilateral in which each pair of opposite sides is parallel and equal in length. Writing in Math Explain how to find the area of a parallelogram with a base of 9 inches and a height of 10 inches.

See TWE margin.

Spiral Review 25

FITNESS A trampoline has a mat that is 10 feet wide and 14 feet long. What is the area of the trampoline mat? (Lesson 3-1, p. 4)


140 ft2 26

Draw a figure that has an area of 18 square units. (Lesson 2-3, p. 71)

Sample answer:

Convert using a table. 27

28

(Lesson 1-5, p. 33)

3 min = 180 s minutes

1

2

3

seconds

60

120

180

days

1

2

3

4

hours

24

48

72

96

96 h =

4

d

Lesson 3-2 Area of a Parallelogram

17

Chapter

Progress Check 1

3

(Lessons 3-1 and 3-2)

Draw each figure that has the given area. 3MG1.2, 4MG1.1, 5MG1.1 1

rectangle, 28 cm2

Sample answer:

2

parallelogram, 60 cm2

Sample answer:

AK

AK AK

AK

Find the area of each rectangle. 4MG1.1 3

4 N ZE N

A=

25 m2

ZE 2 A = 108 yd

Find the area of each parallelogram. 4MG1.1; 5MG1.1 6 DN

GU DN GU

A=

70 cm2

A=

66 ft2

Solve. 4MG1.1 7

DESIGN Lena hung a rectangular mirror on her bathroom wall. The mirror was 85 centimeters high and 67 centimeters wide. What 5,695 cm2 was the area of Lena’s mirror?

8

FOOD Lamont decorated cookies shaped like parallelograms. Each cookie was 84 millimeters tall and 61 millimeters at the base. 5,124 mm2 What was the area of each cookie?

18



5

Lesson

3-3 Area of a Triangle 3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 5MG1.1 Derive and use the formula for area of a triangle and of a parallelogram by comparing each with the formula for the area of a rectangle.

KEY Concept

triangle

VOCABULARY

Cut the parallelogram along the dashed line. Notice you now have two triangles, each one-half the size of the parallelogram.

area the number of square units needed to cover a region or a plane figure (Lesson 2-3, p. 71)

square unit a unit for measuring area I

(Lesson 2-3, p. 71)

I

C

C

h is the height.

_

A=1×b×h 2

A=b×h


A is the area of the triangle.

b is the length of the base.

The location of the height of a triangle can vary. There are three possibilities.

I

I C

C

The height is one side of a triangle.

The height is inside the triangle.

rectangle a quadrilateral with four right angles; opposite sides are equal and parallel and equal in length (Lesson 3-1, p. 4) parallelogram a quadrilateral in which each pair of opposite sides is parallel and equal in length (Lesson 3-2, p. 11)

triangle a polygon with three sides and three angles

I C

The height is outside the triangle.

GO ON Lesson 3-3 Area of a Triangle

19

Example 1

YOUR TURN!

What is the area of the triangle?


JO

N

JO

1. The base of the triangle is 6 inches, and the height of the triangle is 4 inches. 2. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1 × 6 in. × 4 in. A = __ 2 3. Multiply to find the area of the triangle. A = 12 in2 The area of the triangle is 12 square inches.

N

1. The base of the triangle is 8 and the height of the triangle is meters.

meters,

7

2. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1× 8 m× 7 m A = __ 2 3. Multiply to find the area of the triangle. A=

28

m2

The area of the triangle is meters.

28

square

What is the area of the triangle? 1. The base of the triangle has endpoints at (2, 1) and (5, 1). It is 5 - 2 or 3 units long.

2. The height of the triangle has endpoints at (1, 5) and (1, 1). It is 5 - 1 or 4 units long. 3. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1×3×4 A = __ 2 4. Multiply to find the area of the triangle. A=6 The area of the triangle is 6 square units, or 6 units2. 20



Example 2

YOUR TURN! What is the area of the triangle? 1. The base of the triangle has endpoints at (1, 2) and (7, 2) . It is

6

units long.

2. The height of the triangle has endpoints at (4, 2) and

(4, 4) . Its height is

2

units long.

3. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1× 6 × 2 A = __ 2 4. Multiply to find the area of the triangle. A=

6

The area of the triangle is

6

square units, or

6

units2.

Who is Correct?



Chen

1 __ cm2 A = (9 + 16) = 12.5 2

Darby

Amelia

2 A = 9 × 16 = 144 cm

1 cm2 __ A = 2 × 9 × 16 = 72

DN

Circle correct answer(s). Cross out incorrect answer(s). DN

Guided Practice Draw a triangle that has the given area. 1

15 units2

Sample answer:

2

100 units2

Sample answer:


21

Step by Step Practice What is the area of the triangle? 3

The base of the triangle is 5 the triangle is 16 miles.

Step 1

miles, and the height of

Step 2 Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1 × 5 mi × A = __ 2

NJ

16

mi

Step 3 Multiply to find the area of the triangle. A=

40

mi2


40

NJ

square miles.

Find the area of each triangle. 4

The base of the triangle is triangle is

10

N

m


35

35

44

N

m2

1×b×h A = __ 2 1 × 8 ft × A = __ 2 A=

11

square meters.

ft

ft2


GU

44

square feet.

GU

22


10

A=

meters, and the height of the

meters.

1×b×h A = __ 2 1 A = __ × 7 m × 2

5

7


6


45

7

square inches.

The area of the triangle is square units.

20

JO

JO

SLGRQ SLGRQ


Problem-Solving Strategies

Solve. 8

VOLUNTEERING Pam earned a club patch for her jacket by volunteering last week. The patch is shaped like a triangle. It is 66 millimeters tall and has a base of 53 millimeters. What is the area of Pam’s patch? Understand

Draw a diagram. Look for a pattern. Guess and check. ✓ Use a formula. Solve a simpler problem.

Read the problem. Write what you know. The base of the patch is 53 millimeters, and the height of the patch is 66 millimeters.


Plan

Pick a strategy. One strategy is to use a formula. Substitute values of the base and height into the formula for the area of a triangle.

Solve

Use the formula. 1×b×h A = __ 2 1 × 53 mm × A = __ 2 A=

1,749

mm

mm2

The area of the patch is millimeters. Check

66

1,749

square

Use a calculator to check your multiplication.


23

9

FLAGS The Coleman Camp flag is raised every morning. The flag is in the shape of a triangle. It is 36 inches long at its base and has a 2 height of 49 inches. What is the area of the camp flag? 882 in Check off each step.

10

✔

Understand

✔

Plan

✔

Solve

✔

Check

COSTUMES Manuel bought a triangular-shaped bandanna for his costume in the school play. The bandanna is 62 centimeters tall and has a base of 67 centimeters. What is the area of the bandanna?

2,077 cm2 11

Compare the area of a triangle with a base of 12 feet and a height of 6 feet to the area of a parallelogram with a base of 12 feet and a height of 6 feet.

See TWE margin.

Draw a triangle that has the given area. 12

12 units2 Sample answer:

24


13

6 units2

Sample answer:




The area of the triangle is 1,000 square

15

millimeters, or 1,000 mm2.

The area of the triangle is yards, or

63

square

yd2.

NN

ZE

NN

16

63


ZE

20

units2.

17


9

units2.

SLGRQ SLGRQ SLGRQ

SLGRQ


Solve. 18

FOOD For a picnic, Wayne sliced cheese in the shape of triangles. Each slice was 48 millimeters at the base and 51 millimeters tall. What was the area of each slice of cheese?

1,224 mm2 19

HORSES Earl searched for a missing horse near his uncle’s ranch. He searched in an area that was shaped like a triangle. Refer to the photo caption at the right. What was the area of the piece of land that Earl searched?

306,556 yd2

HORSES The area in which Earl searched for the horse had a base of 886 yards and a height of 692 yards.

Vocabulary Check Write the vocabulary word that completes each sentence. 20

21

Area

is the number of square units needed to cover the inside of a region or a plane figure. A(n) triangle is a polygon with three sides and three angles.

GO ON

Lesson 3-3 Area of a Triangle Jerry Irwin/Photo Researchers

25

22

Writing in Math Explain how the area of a triangle is related to the area of a rectangle.

Answers will vary. Sample answer: A rectangle can be cut into two equal triangles. So, the area of a triangle is half the area of a rectangle.

Spiral Review 23

WEATHER Zing turned on the weather channel. She saw a region that had a tornado watch. The region was shaped like a parallelogram and measured 82 miles across at the base and 56 miles high. What was the area of the tornado watch? (Lesson 3-2, p. 11)

4,592 mi2 24

Find the area of the figure. (Lesson 2-3, p. 71)

SLGRQ

SLGRQ

20


25

DN

2 inches

JO

JO

26

(Lesson 2-1, p. 56)

7 centimeters DN

26


Draw a line segment of each length.

square units.


Lesson

3-4 Surface Area of Rectangular Solids KEY Concept Ű TJEF

I

X

CBDL

I

CPUUPN

GSPOU

UPQ

X

3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 4MG1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2). 5MG1.2 Construct a cube and rectangular box from twodimensional patterns to compute the surface area for these objects.

VOCABULARY TJEF

The net can be folded to make a rectangular prism.

surface area the area of the surface of a three-dimensional figure net a flat pattern that can be folded to make a threedimensional figure

I X

Ű

The surface area of a rectangular prism is the sum of the areas of all the faces of the figure. Surface area is measured in square units.

face the flat part of a threedimensional figure that is considered one of the sides square unit a unit for measuring area


A rectangular prism has six faces.

(Lesson 2-3, p. 71)

Example 1 What is the surface area of the rectangular prism? 4. Find the area of faces C and E.

1. Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F.

A=×w A = 2 × 6 = 12

2. Find the area of faces A and F.

5. Find the sum of all the areas of all the faces.

A=×w A = 2 × 5 = 10 3. Find the area of faces B and D.

10 + 30 + 12 + 10 + 30 + 12 = 104 !

"

#

$

The surface area of the rectangular prism is 104 square units.

A=×w A = 5 × 6 = 30

%

GO ON

Lesson 3-4 Surface Area of Rectangular Solids

27

YOUR TURN! What is the surface area of the rectangular prism? 1. Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F. 2. Find the area of faces A and F. A=×w A= 3 ×

8

24

=

3. Find the area of faces B and D. A=×w A= 8 ×

7

56

=

!

"

#

$

4. Find the area of faces C and E. A=×w A= 3 ×

7

%

21

=

5. Find the sum of the areas of all the faces.

24

+

56

+

21

+

24

+

21

56

+

= 202


Example 2

YOUR TURN! What is the surface area of the cube?

What is the surface area of the cube?

1. Draw a net of the cube. 2. Find the area of each face.

!

"

#

$

%

A=×w A = 4 × 4 = 16 3. There are six faces on the cube. Find the sum of the areas of all six faces. A B C D E F 16 + 16 + 16 + 16 + 16 + 16 = 96 The surface area of the cube is 96 square units. 28


!

2. Find the area of each face. A=×w A= 3 ×

3

"

#

$

%

=

9

3. Find the sum of the areas of all six faces.

9 9

+ +

9 9

+ =

9 54

+

The surface area of the cube is square units.

9 54

+


1. Draw a net of the cube.

Who is Correct? What is the surface area of the rectangular prism?

Jermaine

Chad 2

A = 3 × 3 = 9 units 2 A = 5 × 5 = 25 units 2 A = 9 × 9 = 81 units A = 9 + 25 + 81 2 = 115 units

Suja 2

A = 3 × 9 = 21 units 2 A = 3 × 5 = 15 units

2 A = 3 × 9 = 27 units 2 A = 3 × 5 = 15 units

2 A = 5 × 9 = 45 units A = 21 + 15 + 45 + 21 + 15 + 45 2 = 162 units

2 A = 5 × 9 = 45 units A = 27 + 15 + 45 + 27 + 15 + 45 2 = 174 units


Guided Practice Draw a net for a rectangular prism with the given length, width, and height. 2 × 8 × 9 Sample answer:

2

5 × 4 × 7 Sample answer:


1

GO ON Lesson 3-4 Surface Area of Rectangular Solids

29

Step by Step Practice What is the surface area of the rectangular prism? 3

Step 1 Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F.

!

"

Step 2 Find the area of faces A and F. A=×w A= 2 ×

4

=

#

$

%

8

Step 3 Find the area of faces B and D. A=×w A= 4 ×

3

=

12

Step 4 Find the area of faces C and E. A=×w A= 2 ×

3

=

6

Step 5 Find the sum of the areas of all the faces.

8

12

+

6

The surface area of the rectangular prism is

52

square units.

+

12

+

6

+

8

+

52

=

Sample answer:

4

Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F. Follow the steps at the top of page 31 to find the surface area.

30


!

"

%

#

$


Find the surface area of each rectangular prism.

Find the area of faces A and F.

Find the area of faces B and D.

A=×w A=

10

A=×w ×

5

=

50

A=

5

×

7

=

35

Find the area of faces C and E. A=×w A=

10

×

7

=

70

Find the sum of the area of all the faces.

50

+

A

35

+

B

70

+

C

35

70

+

D

50

+

E

= 310

F

The surface area of the rectangular prism is 310 square units. Find the area of faces A and F.

5

A=×w

9

A=

4

×

=

36

Find the area of faces B and D. A=ℓ×w

4

A=

6

×

=

24

Find the area of faces C and E. Copyright © by The McGraw-Hill Companies, Inc.

A=ℓ×w

6

A=

9

×

=

54

Find the sum of the area of all the faces.

36

+

24

+

54

+

36

+

24

54

+

= 228

The surface area of the rectangular prism is 228 square units. 6


7



31


Problem-Solving Strategies ✓ Draw a diagram.

Solve. 8

Look for a pattern. Guess and check. Solve a simpler problem. Work backward.

PRESENTS Selena wrapped a present that was shaped like a cube. Each side measured 9 inches. What is the least amount of wrapping paper that Selena could have used to wrap the present? Understand

Read the problem. Write what you know. Each side of the present is 9 inches.

Plan

Pick a strategy. One strategy is to draw a diagram. Draw a net of the cube.

!

"

#

$

%

Find the area of each face. A=×w A= 9 ×

9

=

81

Find the sum of the areas of all the faces.

81 81

+ 81 + = 486

81

+

81

+

81

+

Selena used at least 486 square inches of wrapping paper. Check

32

Use a calculator to check your multiplicaton and addition.



Solve

9

DESIGN Eddie put carpet on the floor, ceiling, and all of the walls in his video room. The video room has a length of 12 feet, a width of 15 feet, and a height of 10 feet. How much 2 carpet did Eddie use for his video room? 900 ft Check off each step.

10

✔

Understand

✔

Plan

✔

Solve

✔

Check

GEOMETRY What is the surface area of a number cube that has 15 millimeter edges?

1,350 mm2 11

Use what you know about how to find the area of triangles and rectangles to find the surface area of this triangular prism. (Hint: This triangular prism has 2 sides that are triangles and 3 sides that are rectangles.)

JO JO JO JO

See TWE margin.


JO

Skills, Concepts, and Problem Solving Draw a net for a rectangular prism with the given length, width, and height. 12


13



33

Find the surface area of each rectangular prism. 14

The surface area of the rectangular 168 prism is square units.

15


16


17


Solve. 18

JEWELRY Daisy made the jewelry box shown. What is the surface area of the jewelry box?

1,944 cm2 ART Quinn decorated a rectangular-shaped chest with wallpaper. The length of the chest is 3 feet, the width is 5 feet, and the height is 2 feet. What is the least amount of wallpaper Quinn used?

62 ft2 Vocabulary Check Write the vocabulary word that completes each sentence. 20

21

22

net A(n) is a flat pattern that can be folded to make a three-dimensional figure. A(n) figure.

face

Surface area

is the flat side of a three-dimensional is the area of the surface of a three-

dimensional figure.

34 Corbis


JEWELRY Daisy’s jewelry box is shaped like a cube. The sides measure 18 centimeters each.


19

23

Writing in Math Explain how to find the surface area of a rectangular prism.

Sample answer: To find the surface area of a rectangular prism, find the sum of the areas of all the faces of the rectangular prism.

Spiral Review 24

TABLES The top of a table measures 54 inches by 116 inches. What 6,264 in2 is the area of the top of the table? (Lesson 3-1, p. 4)

25

SCHOOL PROJECT Larry made a poster for a school project. He wants to make a big envelope to protect the poster when he carries it to school. The poster is 24 inches by 18 inches. Larry needs two identical sheets of plastic that he will tack together on three sides. What is the area of one of the sheets Larry needs?

432 square inches 26

DECORATING Belinda plans to paint one wall of her bedroom bright blue. She needs to know the area of the wall to decide how much paint she needs. The wall is 14 feet by 8 feet. What is the area of the wall Belinda will paint?


112 square feet 27

Draw a figure that has a perimeter of 20 centimeters. (Lesson 2-2, p. 63)

Sample answer: DN

DN

DN

DN

DN

Lesson 3-4 Surface Area of Rectangular Solids

35

Chapter

Progress Check 2

3


Draw a triangle that has the given area. 3MG1.2, 5MG1.1 1


2

Sample answer:

4 units2

Draw a net for a rectangular prism with the given length, width, and height. 3MG1.2 3


4


5

The surface area of the rectangular prism is square units.

268

6

The surface area of the rectangular prism is square units.

178


Find the surface area of each rectangular prism. 3MG1.2, 4MG1.1

Solve. 3MG1.2, 4MG1.1, 5MG1.1 7

SIGNS Stella designed the triangular-shaped sign shown at the right for a travel agency. What is the area of Stella’s sign?

IN

434 in2 8

STORAGE A storage cabinet is 24 inches wide, 26 inches long, and 40 inches high. What is the surface area of the storage cabinet?

5,248 in2 36


IN

Lesson

3-5 Volume of Rectangular Solids KEY Concept The amount of space inside a threedimensional figure is the volume of the figure.

I

The volume of a rectangular solid is the product of its length, width, and height. V is the volume of the solid figure.

V=×w×h is the length.

or

X

Ű

w is the width.

VOCABULARY volume the number of cubic units needed to fill a threedimensional figure or solid figure

V = wh

(Lesson 2-4, p. 77)

cube a three-dimensional figure with six congruent square faces (Lesson 2-4, p. 77)

h is the height.

cubic unit a unit for measuring volume (Lesson 2-4, p. 77)

Volume is measured in cubic units.


3MG1.2 Estimate or determine the area and volume of solid figures by covering them with squares or by counting the number of cubes that would fill them. 5MG1.3 Understand the concept of volume and use the appropriate units in common measuring systems to compute the volume of rectangular solids.

Example 1

YOUR TURN!

What is the volume of the rectangular prism?

What is the volume of the rectangular prism?

1. The length of the cube is 3 units. The width of the cube is 7 units. The height of the cube is 4 units. 2. Substitute the length, width, and height into the volume formula. V=×w×h V=3×7×4 3. Multiply. V = 84 The volume of the rectangular prism is 84 cubic units.

1. The length of the rectangular prism is 5 units. The width of the rectangular prism is 4 units. The height of the rectangular prism is 8 units. 2. Substitute the length, width, and height into the volume formula. V=×w×h V= 5 ×

4

×

8

3. Multiply. V = 160 The volume of the rectangular prism is 160 cubic units.

GO ON

Lesson 3-5 Volume of Rectangular Solids

37

Example 2

YOUR TURN!

What is the volume of the cube?

What is the volume of the cube? 1. The length of the cube is 6 units.

1. The length of the cube is 5 units.

The width of the cube is 6 units.

The width of the cube is 5 units. The height of the cube is 5 units. 2. Substitute the length, width, and height into the volume formula.

The height of the cube is

units.

2. Substitute the length, width, and height into the volume formula. V=×w×h V= 6 ×

V=×w×h V=5×5×5 V = 53

6

V=

6

×

6

63

3. Multiply.

3. Multiply.

V = 216

V = 125 The volume of the cube is 125 cubic units.

The volume of the cube is 216 cubic units.

Who is Correct? What is the volume of the rectangular prism?

V=7×9×5 = 315 cubic units

Kristen V = 63 + 45 + 35 + 63 + 45 + 35 = 286 cubic units


38


Jamie V=7×9×5 = 385 cubic units


Ramiro

Check your answer. Remember, you can find the volume of a solid figure by counting the number of cubic units it contains.

Guided Practice 1

How many cubes are in this rectangular prism?

8

2

How many cubes are in this rectangular prism?

35

Step by Step Practice Find the volume of the rectangular prism. 3

Step 1 The length of the rectangular prism is

4

units.

The width of the rectangular prism is

8

units.

The height of the rectangular prism is

2

units.

Step 2 Substitute the length, width, and height into the volume formula. V=×w×h


V=

4

×

8

×

2

Step 3 Multiply. V=

64

The volume of the rectangular prism is

64

cubic units.

Find the volume of each rectangular prism. 4

Substitute the length, width, and height into the volume formula. Then multiply. V=×w×h V=

6

×

10

×

3

V = 180 The volume of the rectangular prism is 180 cubic units.

GO ON

Lesson 3-5 Volume of Rectangular Solids

39

5

V=×w×h V=

2

×

12

×

5

V = 120 The volume of the rectangular prism is 120 cubic units.

6

The volume of the rectangular prism is 288 cubic units.

7



Problem-Solving Strategies ✓ Use a model.

Solve. 8

Look for a pattern. Guess and check. Act it out. Work backward.

HOUSES Hillary has a window air conditioner in her bedroom that is 14 inches long, 17 inches wide, and 15 inches tall. What is the volume of Hillary’s air conditioner? Understand

Read the problem. Write what you know.

a width of

15 Plan

17

14

inches,

inches, and a height of

inches.

Pick a strategy. One strategy is to use a model. Stack cubes to model the air conditioner.

Solve

Use the formula. V=×w×h V=

14

in. ×

17

in. ×

15

in.

V = 3,570 in3 The volume of Hillary’s air conditioner is 3,570 cubic inches. Check

40

Use a calculator to check your multiplication.



The air conditioner has a length of

9

CONSTRUCTION Yamin’s family has a storage shed that is 4.5 yards wide, 28 yards long, and 4.5 yards high. What is the 3 volume of the shed? 567 yd Check off each step.

10

11

✔

Understand

✔

Plan

✔

Solve

✔

Check

PACKAGING Mrs. Reynolds put together a card box for her daughter’s graduation party. The card box was shaped like a cube. Each side measured 48 centimeters. What was the 110,592 cm3 volume of the card box? Compare the volume of the rectangular prism shown at the right to its surface area.

See TWE margin.

Skills, Concepts, and Problem Solving Copyright © by The McGraw-Hill Companies, Inc.

Find the volume of each rectangular prism.

30

12


13


14


cubic units.

15


GO ON Lesson 3-5 Volume of Rectangular Solids

41

Solve. 16

CONSTRUCTION Marco’s father built the tree house shown at the right. What was the volume of the tree house?

420 ft3 17

COLLECTIONS Lydia used a shoe box for her rock collection. The shoe box was 12 inches long, 6 inches wide, and 4.5 inches high. What was the volume of the shoe box?

324 in3 Vocabulary Check Write the vocabulary word that completes each sentence.

CONSTRUCTION The tree house that Marco’s father built had a height of 7 feet, a length of 10 feet, and a width of 6 feet.

18

Cubic unit

19

Volume

is the number of cubic units needed to fill a three-dimensional figure or solid figure.

20

Writing in Math Explain how to find the volume of a rectangular prism.

is a unit for measuring volume.

Sample answer: Find the length, width, and height of the rectangular prism. Substitute the length, width, and height into the formula for the volume of a rectangular prism. Then multiply.

21

MUSIC Marika’s stereo speakers are cubed-shaped. Each side measures 20 millimeters. What is the surface area of each speaker? (Lesson 3-4, p. 27)

2,400 mm2 22

Draw a triangle that has an area of 320 square units. (Lesson 3-3, p. 19)

Sample answer:

42 Corbis



Spiral Review

Chapter

Study Guide

3

Vocabulary and Concept Check face, p. 27

Write the vocabulary word that completes each sentence.

net, p. 27

1

parallelogram, p. 11 2

rectangle, p. 4 square, p. 4

3

surface area, p. 27 triangle, p. 19

4

5

A(n)

square is a rectangle with four congruent sides.

net A(n) is a flat pattern that can be folded to make a three-dimensional figure. parallelogram A(n) is a quadrilateral in which each pair of opposite sides is parallel and equal in length. rectangle A(n) right angles.

is a quadrilateral with four

Surface area

is the area of the surface of a three-dimensional figure.


Label each diagram below. Write the correct vocabulary term in each blank. 6

triangle

8

The net shown is of a

!

"

7

#

cube

and has

6

face

faces.

$

%

Chapter 3 Study Guide

43

Lesson Review

3-1

Area of a Rectangle

(pp. 4–10)

Find the area of each rectangle. 9

DR

Example 1 Find the area of the rectangle using the formula A = × w.

GLAFCQ GLAFCQ

is the length of the rectangle. A is the area of the rectangle.

DR 2 A = 99 square feet or 99 ft

10

w is the width of the rectangle.

A=×w

The length of the rectangle is 6 inches, and the width is 3 inches. Substitute these values into the formula. Multiply.

AK

AK

A=×w A = 6 in. × 3 in. The area of the rectangle is 18 square inches. A = 18 in2

A = 120 square centimeters or

120 cm2

Find the area of each square.

What is the area of the square? K WB

K 2 A = 49 square meters or 49 m

WB

The length of the square is 5 yards, and the width is 5 yards.

12

Substitute these values into the formula. Multiply. AK

AK

A = 100 square centimeters

or 100 cm2

44


A=l×w A = 5 yd × 5 yd The area of the square is 25 square yards. A = 25 yd2


11

Example 2

3-2

Area of a Parallelogram

(pp. 11–17)

Find the area of each parallelogram.

Example 3 What is the area of the parallelogram?

13 AK

GL

AK

A = 75 square centimeters

or 75 cm2

The base of the parallelogram is 12 inches, and the height is 10 inches.

14

Substitute these values into the formula. Multiply.

DR DR 2 A = 24 square feet or 24 ft

3-3

Area of a Triangle

(pp. 19–26)

Find the area of each triangle. Copyright © by The McGraw-Hill Companies, Inc.

GL

15 DR DR

2 A = 1.5 square feet, or 1.5 ft

16

AK

AK

A = 99 square centimeters, or 99 cm2

A=b×h A = 12 in. × 10 in. A = 120 in2

The area of the parallelogram is 120 square inches.

Example 4 What is the area of the triangle? The base of the triangle is 10 inches, and the height of the triangle is 16 inches. Substitute values of the base and height into the area of a triangle formula. 1×b×h A = __ 2 1 × 10 in. × 16 in. A = __ 2

GL

GL

Multiply to find the area of the triangle. A = 80 in2 The area of the triangle is 80 square inches.


45

3-4

Surface Area of Rectangular Solids

(pp. 27–35)

Find the surface area of each rectangular prism.

Example 5 What is the surface area of the rectangular prism? Draw a net of the rectangular prism. Label the faces A, B, C, D, E, and F.

17

Find the area of faces A and F. The surface area of the cube is 216 square units.

A=l×w A=2×4=8

!

Find the area of faces B and D.

18

"

#

$

%

A=l×w A = 4 × 5 = 20 Find the area of faces C and E. A=l×w A = 2 × 5 = 10 Find the sum of the areas of all the faces. The surface area of the cube is 486 square units.

8 + 20 + 10 + 8 + 20 + 10 = 76

Find the surface area of each rectangular prism. 19

20

The surface area of the rectangular prism is 248 square units. The surface area of the rectangular prism is 150 square units.

46




3-5

Volume of Rectangular Solids

(pp. 37–42)


Example 6 Find the volume of the rectangular solid using the formula below:

21

is the length.


V=×w×h V is the volume of the solid figure.

22

h is the height.

w is the width.

The length of the cube is 4 units. The width of the cube is 9 units. The volume of the rectangular prism is 135 cubic centimeters.

The height of the cube is 2 units. Substitute the length, width, and height into the volume formula.

23


V=×w×h V=4×9×2 V = 72 The volume of the rectangular prism is 216 cubic units.


24



47

Chapter

Chapter Test

3

Find the area of each rectangle. 4MG1.1 1

2 AK

GL AK GL

A=

56 in2

90 cm2

A=

Draw a rectangle that has the given area. 3MG1.2 3


4

49 units2

Sample answer:

Find the area of each parallelogram. 5MG1.1 6


5 ZE ZE

A=

JO

2

60 yd

JO

A=

792 in2

Find the area of each triangle. 3MG1.2, 4MG1.1, 5MG1.1 7

8

DN

DN 2 A = 6 units

48

Chapter 3 Test

A=

81 cm2

GO ON

Find the surface area of each rectangular prism. 3MG1.2, 4MG1.1 9

The surface area of the rectangular prism is units.

122

square

10

The surface area of the rectangular prism is units.

234

square

Find the volume of each rectangular prism. 3MG1.2 11


180

cubic units.

12


168

cubic units.


Solve. 4MG1.2, 5MG1.1 13

PETS The floor of Jet’s doghouse is a rectangle with a length of 32 inches and a width of 20 inches. What is the area of the floor of 2 Jet’s doghouse? 640 in

14

BAKING Betsy made dough cutouts in the form of triangles with a base of 11 centimeters and a height of 8 centimeters. 2 What was the area of each triangular dough cutout? 44 cm

Correct the mistakes. 15

Margaret bought a rug that was a square with 8 feet on each side. Later that day Margaret e-mailed her sister to describe the rug. What mistake did Margaret make?

IDN

CDN

'SPN.BSHBSFU 4VCKFDU/FXSVH 5P4JTUFS

.YNEWRUGISSQUAREFEET FTSQUAREFEET

.ARGARET

Sample answer: The mistake Margaret made was that she found the perimeter of the rug rather than the area. 16

Show how Margaret should have found the area of the rug.

Sample answer: To find the area of the rug, you multiply the length and width. 8 × 8 = 64 square feet, or 64 ft2. Chapter 3 Test

49

Chapter

Standards Practice

3

Choose the best answer and fill in the corresponding circle on the sheet at right. What is the area of the rectangle?

1

4MG1.1

4

N

N

Malik has a parallelogram-shaped mouse pad. It has a base length of 10 inches and a height of 8 inches. What is the area of the mouse pad?

5MG1.1

A 23 m2

F 9 in2

H 36 in2

G 18 in2

J 80 in2

B 46 m2 C 90 m2 5

D 100 m2

Use the formula for the area of a rectangle to find the area of one of the triangles in the figure. 5MG1.1

One wall in Mr. Avsharian’s classroom measures 35 feet by 14 feet. What is the area of this wall? 4MG1.1

2

JO

F 49 ft2 G 98 ft2

JO

A 16 in2

C 32 in2

J 560 ft2

B 24 in2

D 64 in2

Find the area of each figure. Which sentence is true? 5MG1.1

3

'JHVSF"

6

What is the area of the right triangle?

5MG1.1

'JHVSF#

DN

DN DN

LN

DN

A Area A > Area B B Area B > Area A C Area A < Area B D Area A = Area B

50

Chapter 3 Standards Practice

LN

F 21 km

H 54 km2

G 54 km

J 108 km2 GO ON


H 490 ft2

7

What is the perimeter of the rectangle?

10

3MG1.3

What is the area of the shaded figure?

3MG1.2 ZE ZE

F 7 square units

A 48 yards

G 8 square units

B 48 yd2

H 21 square units

C 63 yards

J 28 square units

D 63 yd2

ANSWER SHEET 8

Alyssa has a closed shoe box that measures 10 inches by 7 inches by 5 inches. What is the volume of the shoe box? 3MG1.2 F 22 in3


G 350 in

9

H 350 in2 3

J 350 in

What is the surface area of the rectangular solid? 3MG1.2

ZE

Directions: Fill in the circle of each correct answer. 1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

ZE ZE

A 18 yd2 B 186 yd2 C 216 yd2

Success Strategy Try to answer every question. Work out the problem and eliminate answers you know are wrong. Do not change your answers unless you are very uncertain about your first answer choice.

D 248 yd2


51

Chapter

4

Angles and Circles Angles, circles, and shapes surround us. We must understand and measure angles, circles, and other shapes to build and use things like buildings, planes, maps, rockets, cars, and bicycles.


52

Chapter 4 Angles and Circles

Getty Images

STEP

STEP

1 Quiz

Are you ready for Chapter 4? Take the Online Readiness Quiz at ca.mathtriumphs.com to find out.

2 Preview

Get ready for Chapter 4. Review these skills and compare them with what you’ll learn in this chapter.

What You Know

What You Will Learn

You know that if you do a 180° turn, that you turn to face the opposite way.

Lesson 4-2 Angles are figures that are often measured in degrees. ¡

This angle measures 180°. It is a straight angle. You know how to describe and recognize some figures. TRY IT!


2

Triangles and quadrilaterals can be classified by their angles and sides.

See TWE margin.

A parallelogram has four sides. Each pair of opposite sides is parallel and equal in length.

See TWE margin.

An equilateral triangle has three sides equal in length and angles that are the same size.

Describe the figures. 1

Lesson 4-3

You know how to compare numbers.

Lesson 4-5

Example: 6 = 6

When two figures have the same size and shape, they are congruent .

TRY IT!

6

Use >, 9

6

1,234 = 1,234

)

¡

4

¡

¡

¡

5

'

¡

¡

(

These two triangles are congruent.

53 Corbis

Lesson

4-1 Lines 5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools.

KEY Concept ;

GH IJ

point Z A point is named using capital letters and is represented by dots. "

# N

$

%

, .

'

ray EF or EF

+

/

KL intersects MN. Intersecting lines meet or cross at a point. This symbol indicates a right angle.

0

2

1

3

OP ⊥ QR Perpendicular lines meet, or intersect, to form right angles . ⊥ means is perpendicular to.

You should know the meaning of the terms and their symbols.


point an exact location in space line a set of points that goes straight in opposite directions without ending line segment a part of a line that has two endpoints ray

A ray is named by its endpoint and a point on the ray. The endpoint of the ray is listed first.

54

VOCABULARY

a part of a line that has one endpoint and extends indefinitely in one direction parallel lines lines that are the same distance apart; parallel lines do not meet intersecting lines lines that meet or cross each other perpendicular lines lines that meet or cross each other to form right angles right angle an angle with a measure of 90°


&

)

-

line segment −− CD or CD A line segment is named by its endpoints.

*

Parallel lines do not intersect. means is parallel to.

line AB or AB line m A line is named by two of the points on the line or by a single letter.

(

Example 1

YOUR TURN!

Draw a line. Name the line QR .

1. Use a ruler to draw a line.

1. Use a ruler to draw a line. 2. Draw arrows on each end of the line to show that the line does not end in that direction. 3. Place two points on the line. Label the points Q and R. 2

Draw a ray. Name the ray ST .

3

one end of the 2. Draw an arrow on line to show that the ray does not end in that direction. point on the other end. 3. Place a S Label the point because the endpoint of the ray is listed first. point

4. Draw a

Label the point

close to the arrow.

T

4

Example 2

. 5

YOUR TURN! Name each line. Identify the relationships.

Name each line. Identify the relationships. .

8 0


:

9

/

;

. 1. One line is WX . The other line is YZ 2. The lines cross, or intersect. 3. The lines intersect to form right angles. So, the lines are perpendicular lines. 4. Write the notation for the perpendicular lines.

1

MN . 1. One line is OP The other line is

.

2. The lines intersect . 3. Do the lines intersect to form right no angles? The lines are 4.

MN

intersecting

intersects

OP

lines. .

⊥ YZ WX

GO ON Lesson 4-1 Lines

55

Example 3

YOUR TURN!

Draw three lines GH , KN is , and TW . GH parallel to KN . and perpendicular to TW 5

(

Draw three lines AB , FH , and ZY . AB is parallel to ZY and perpendicular to FH .

,

;

"

)

'

)

/ #

8

. 1. Draw GH

. 1. Draw AB

2. Draw KN so that every point is equal . distance from line GH

2. Draw

so that it forms a right angle 3. Draw TW . with GH

3. Draw

:

ZY so that every point is equal . AB distance from line FH so that it forms a right . AB angle with

is also perpendicular Notice that TW to KN .

ZY is also perpendicular to FH .

Who is Correct? Identify the figure. 6

7

Mick

Erina

There are two endpoints. It is a ray. UV

There are no endpoints. It is a line. UV

There are two is a endpoints. It−− UV segment.


Mira


Guided Practice Draw each figure. 1

and UT intersecting lines WV

2

BC #

8 6

5

7

56


$


Name each line. Identify the relationships. '

)

(

*

FG

Step 1 Name the lines.

HI

;

Step 2 The lines do not intersect . The lines are

parallel

lines.

FG HI

Step 3 Write the notation for the lines.

Name each line. Identify the relationships. 4

AB

The lines are

and

YZ

.

The lines intersect . The lines are AB


5

⊥

perpendicular

" #

:

;

lines.

YZ

DE

The lines are

and

JK

.

The lines intersect .

,

+

intersecting

The lines are

%

&

lines.

DE intersects JK 6

MN

The lines are

and

parallel

The lines are

ST

.

.

4

/

5

lines.

ST MN 7

The lines are

WX

The lines are

perpendicular

⊥ YZ WX

and

YZ

8

.

lines.

:

9

;

GO ON Lesson 4-1 Lines

57

Draw each relationship described. 8

QR NO

/

0

2

3

$

⊥ KL CD

9

,

%


Problem-Solving Strategies ✓ Use a diagram.

Solve. 10

Look for a pattern. Act it out. Solve a simpler problem. Work backward.

MAPS Pia lives on Maple Street, and her aunt lives on Elm Street. On the map, what type of lines do the two streets form? Understand

Read the problem. Write what you know. The streets in the problem are Maple Street and Elm Street .

&MN4U

intersect

.BQMF4U

" WF SD I

The streets

#J

Solve

.

Do the streets intersect to form right angles? So the streets are

lines.

MAPS Keisha walked her dog from her house to the park. Ursula walked from her house to the library. Draw lines for the paths that Keisha and Ursula took. What type of lines do the intersecting two paths form? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check


,FJTIBT IPVTF -JCSBSZ

6STVMBT IPVTF

1BSL


58

perpendicular

yes

Use a protractor to measure the angle between Maple Street and Elm Street. Do the streets intersect yes to form 90° angles?

Check

11

U

Pick a strategy. One strategy is to use a diagram. Look at the diagram. Do the streets intersect?

L4

0B

Plan

12

DESIGN Mr. Inez bought bunk beds for his twin sons. In the picture to the right, what kind of lines are the two bases?

#BTF

parallel Is the pair of lines shown below intersecting lines? Explain.

13

#BTF

Yes; the lines will continue in both directions indefinitely, so the lines will eventually intersect.

Skills, Concepts, and Problem Solving Draw each figure. 14


16

parallel lines AB and FG "

#

'

(

15

-

WY

17

8

line segment LM

:

.

point X 9


The lines are

BC

parallel

The lines are BC 19

and

FG

The lines are

perpendicular

⊥

.

lines.

#

$

&

'

EF

The lines are

FG

EF

and

IJ

'

.

lines.

*

(

+

IJ GO ON Lesson 4-1 Lines

59


TU

The lines are

parallel

The lines are TU 21

and

XY

.

5

6

9

:

lines.

XY

CD

The lines are

and

intersecting

The lines are

WV

$

.

8

lines.

7

%

intersects WV CD 22

ST , CR , and UP .

The lines are

5

CR UP

6

1

ST ⊥ UP $

ST ⊥ CR

23

4

GT , DF , and

The lines are

JS .

24

The lines are

EF

DF ⊥

JS

TS OP

GT ⊥

JS

EF OP

%

TS , and OP .

TS

'

+

EF ,


GT DF

5

3

4 1

&

4

5

' 0

(

Draw each figure. 25 BD MN

#

/

26

⊥ HY AZ

:

" %

60


.

; )

Draw each figure. 27 Parallel lines CD and LM . perpendicular to XY $

28

and VT Parallel lines QR . perpendicular to NO /

-

9

2

:

3

7 %

.

5 0

Solve.

"

29

HOBBIES Kelsey’s Kites designs kites. Classify the lines that perpendicular form the cross section of the kite.

30

CONSTRUCTION Doug has an iron fence that surrounds his house and yard. What type of lines do the vertical

#

$ 9

iron bars form?

" %

parallel


:


Perpendicular

lines are lines that meet or cross each other

to form right angles. 32

ray A(n) is a part of a line that has one endpoint and extends indefinitely in one direction.

33

A (n)

34

point

Intersecting

is an exact location in space. lines are lines that cross or meet at a point.

GO ON Lesson 4-1 Lines Ingram Publishing/SuperStock

61

35

Writing in Math Explain how to draw line segment JK.

Use a ruler to draw a line. Place point endpoints on both ends of the line. Label the points J and K.

Spiral Review 36

PACKAGING The tissue box that Tobias keeps on his desk at work is shaped like a cube. Each edge of the tissue box is 13 centimeters long. What is the volume of the tissue box? (Lesson 3-5, p. 37)

2,197 cm3



(Lesson 3-2, p. 11)

36

square meters or

56

square feet or

36

m2.

N N

38


56

ft2.


GU

GU

Convert.

(Lesson 1-4, p. 25)

39

1.5 T = 3,000 lb

41

3 pt =

62


48

fl oz

40

72 c =

42

2.5 gal =

18

20

qt

pt

Lesson

4-2 Angles 5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools.

KEY Concept " WFSUFY Ȝ#Ȝ

VOCABULARY

#

$

To name an angle, identify the vertex. The vertex of this angle is B. An angle can be named by using the point on one ray, the vertex, and then the point on the other ray, such as ∠ABC or ∠CBA. An angle can also be named using the letter of the vertex, such as ∠B. An angle is sometimes named by a number, such as ∠1. Angles are often measured in degrees. They can be classified, or grouped, according to their measures. You can use a protractor to measure an angle. %

protractor an instrument marked in degrees, used for measuring or drawing angles

degree a unit for measuring angles

&

'

Acute angles measure between 0° and 90°. ∠CEF is an acute angle with a measure of 45°. Right angles measure exactly 90°. ∠DEF is a right angle with a measure of exactly 90°. (

*

)


vertex a point of intersection of the sides of an angle (the plural is vertices)

$

angle a figure formed by two rays with the same endpoint

+

Obtuse angles measure between 90° and 180°. ∠GIJ is an obtuse angle with a measure of 135°. Straight angles measure exactly 180°. ∠HIJ is a straight angle with a measure of exactly 180°. Classify an angle as acute or obtuse to help you decide which scale of the protractor, the inner or outer, gives the angle’s measurement.

GO ON Lesson 4-2 Angles

63

Example 1

YOUR TURN! Identify the type of angle.

Identify the type of angle.

1. The angle does not have a square corner. It is not a right angle.

1. Does the angle look like a right angle?

2. The angle looks greater than 90°. It is an obtuse angle.

less than 2. The angle looks acute angle It is a(n) .

Example 2

YOUR TURN!

&

'

2. Place the center of the protractor at point E. Line up EF with the 0° mark on the protractor.

¡ 9

:

'

:

8


9

. 4. Draw XW

%

'


8

4. Draw ED .

64

3. Use the inner scale. Draw a point W at 75 °

&

&

:

9

'

3. Use the inner scale. Draw a point D at 135°.

&

%

2. Place the center of the protractor at point X . Line up XY with the 0 ° mark on the protractor.

:

9

. 1. Draw XY

90°.

Draw ∠WXY that measures 75°.

Draw ∠DEF that measures 135°. 1. Draw EF .

no

Example 3

YOUR TURN! Measure and identify the angle.

Measure and identify the angle.

1. Place the center of the protractor at the vertex.

9

1. Place the center (hole or ⊥ symbol) of the protractor at the vertex, point Y.

:

;

with the line that extends 2. Line up YZ from 0° to 180° on the protractor.

140°

∠ABC measures

.

∠ABC is a(n)

obtuse

angle.

3. Look at point C. Use the 0° reading to know whether to read the inner scale or the outer scale on the protractor. Read the measure of the angle where BC passes through the inner scale.

with the line that extends 2. Line up AB from 0° to 180° on the protractor.

"

3. Look at point Z. Use the 0° reading to know whether to read the inner scale or the outer scale on the protractor. Read the measure of the angle where YX passes through the inner scale.

#

$

∠XYZ measures 60°. Copyright © by The McGraw-Hill Companies, Inc.

∠XYZ is an acute angle.

Who is Correct? What is the measure of ∠V?

6

Rachel

Ivan

Dion

90°

95°

100°

7

8



65

Guided Practice Identify each type of angle.

acute

1

obtuse

2

Draw an angle with the given measurement. 90°

4

130°

3


Measure and identify the angle. ' Copyright © by The McGraw-Hill Companies, Inc.

(

)

G

Step 1 Place the center of the protractor at point

.

Step 2 Line up the vertex and the line on the protractor that extends from 0° to 180° with Step 3 Read from the

outer

GH

.

scale.

'

(

)


66

30 °. ∠FGH measures ∠FGH is a(n) acute angle.

Step 4 Read the measure of the angle passes through where GF the outer scale.

Measure and identify each angle. 6

Line up the vertex and the line on the protractor that extends

KL

from 0° to 180° with

inner

Read from the

.

+

scale.

KJ

Read the measure of the angle where through the inner scale.

100

∠JKL measures ∠JKL is a(n)

obtuse

passes ,

-

°. +

angle.

,

-

7

∠MNO measures ∠MNO is a(n)

90 right

°. angle.

.


/

8

∠TUV measures ∠TUV is a(n)

85 acute

°. angle.

5

6

9

∠RST measures

180

0

7

°.

∠RST is a(n) straight angle.

3

4

5


67



Solve. 10

Look for a pattern. Guess and check. Solve a simpler problem. Work backward.

BASEBALL While rounding the bases after hitting a home run, Quincy noticed the first and second bases formed an angle. What type of angle is formed by the line from first base to second base and the line from first base to home plate? Understand

Read the problem. Write what you know. Quincy was looking at the angle

first

formed by the line from base to

second base and the line first base

from

home plate

to . TFDPOE

Plan

Pick a strategy. One strategy is to use a diagram.

Solve

Place the center of the protractor on first base. Line up the vertex and the line on the protractor that extends from 0° to 180° with

the line from first base to second base

UIJSE

. IPNF

Read the measure of the angle where passes through the outer scale.

11

68

°.

right

angle.

Review the definition of a

right

The angle is a(n) Check

90

angle.

BILLIARDS How many acute angles does a pool rack have? 3 Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check


(t)Rudi Von Briel/PhotoEdit, (b)Corbis


the line from first base to second base The angle measures

mSTU

12

TIME Justin eats lunch at school at 11:15. What type of angle do the hands on a clock form when the time is 11:15?

obtuse

Ruben said the measure of the angle is 105°. Explain Ruben’s error.

13

Sample answer: The measure of the angle is 75°. Ruben most likely read the outer scale of a protractor.


14

straight

15

obtuse

16

acute

17

right

Draw an angle with the measurement given. 25°

19

90°

21

165°

20

105°

18


Identify each type of angle.


69

Measure and identify each angle. 22

65

∠ABC measures

acute

∠ABC is a(n)

°. "

angle.

#

23

∠WXY measures

180

$

°.

∠WXY is a(n) straight angle.

24

∠PQR measures ∠PQR is a(n)

25

obtuse

∠STU measures ∠STU is a(n)

170

90 right

8

9

:

2

3

°. 1

angle.

°. angle.

4

5

6

Solve. 26

SAILING Val hoisted a sail to prepare for the start of a race. What types of angles are formed by the sides of the sail?

27

CARS After back surgery, Kono could not sit up straight while riding in a car. He had to recline the seat at an angle that measured 110°. What type of angle was Kono’s reclining position?

obtuse

SAILING This sailboat has triangular sails.

Vocabulary Check Write the vocabulary word that completes each sentence.

angle

28

A(n)

29

A(n) protractor is an instrument marked in degrees, used for measuring or drawing angles.

70


Corbis

is formed by two rays with a common endpoint.


acute

30

A(n) obtuse angle is an angle that measures greater than 90° but less than 180°.

31

A(n) acute less than 90°.

32

Writing in Math Explain how to draw ∠XYZ measuring 125°.

angle is an angle that measures greater than 0° but

. Place the center of the protractor at point Y. Line up YZ with the 0° Draw YZ . mark on the protractor. Use the outer scale. Draw a point X at 125°. Draw YX

Spiral Review 33

FENCES Derek’s horse runs in a paddock that is surrounded by a fence. What type of lines do the slats on the fence form?

parallel

(Lesson 4-1, p. 54)



35

(Lesson 3-3, p. 19)

The area of the triangle is 20 square yards or 20

ZE

yd2.

ZE

The area of the triangle is 22, 100 square millimeters or 22, 100 mm2.

FENCES Derek’s horse runs in the paddock surrounded by this fence.

NN

NN

Convert.

(Lesson 1-1, p. 4)

75

36

0.75 m =

38

54 m = 0.054 km

cm

37

0.2 cm = 0.02 dm

39

6.3 dm = 630 mm

Lesson 4-2 Angles Corbis

71

Chapter

Progress Check 1

4


Draw each figure. 5MG2.1 1

⊥ CD AB

2

−− FG '

(

$ "

# %

Draw an angle with the measurement given. 5MG2.1 3

150°

4

40°

¡ ¡

Name each line. Identify the relationships. 5MG2.1 5

4

5

6

7

6

: 8

9 ;

The lines are ST ll UV

parallel

lines.

The lines are perpendicular ⊥ YZ WX

lines.

7

8

%

2

¡ &

'

¡ 3

∠DEF measures ∠DEF is a(n)

50 acute

°. angle.

∠QRS measures ∠QRS is a(n)

4

90 right

Solve. 5MG2.1

°. angle.

n

9

TIME Is the measure of the angle formed by the hands of the clock greater than, less than, or equal to 90°? See TWE margin.

n

72



Measure and identify each angle. 5MG2.1

Lesson

4-3 Triangles and Quadrilaterals 5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools.

KEY Concept Classify Triangles by Angles

VOCABULARY

When you classify, you place items in groups. This symbol indicates a right angle.

congruent line segments that have the same length, or angles that have the same measure

right triangle one 90° angle

obtuse triangle one angle greater than 90° and less than 180°

acute triangle all three angles less than 90°

Classify Triangles by Sides

quadrilateral a polygon that has four sides and four angles trapezoid a quadrilateral with exactly one pair of parallel sides parallelogram a quadrilateral with four sides in which each pair of opposite sides is parallel and congruent

This symbol indicates congruent sides. Copyright © by The McGraw-Hill Companies, Inc.

triangle a polygon with three sides and three angles

rectangle a quadrilateral with four right angles; opposite sides are congruent and parallel

equilateral triangle three congruent sides and three congruent angles

rhombus a parallelogram with four congruent sides

This symbol indicates congruent angles.

isosceles triangle At least two congruent sides and two congruent angles

square a rectangle with four congruent sides

scalene triangle no congruent sides GO ON Lesson 4-3 Triangles and Quadrilaterals

73

Classify Quadrilaterals trapezoid quadrilateral with exactly one pair of parallel sides

This symbol indicates parallel sides.

parallelogram each pair of opposite sides is parallel and congruent

rectangle parallelogram with four right angles

opposite angles of each side are also congruent

square rectangle with four congruent sides

rhombus parallelogram with four congruent sides

Symbols

74

parallel lines

congruent sides

congruent angles

right angle



Figures often have symbols that indicate congruence.

Example 1 Identify the figure.

1. The figure has three sides and three angles. The figure is a triangle. 2. Two sides are equal in length. The figure is an isosceles triangle, which means two angles are congruent. 3. One angle is a right angle. The figure is a right, isosceles triangle.

YOUR TURN! Identify the figure.

1. The figure has 3 sides and angles. The figure is a(n)

triangle

3

.

2. Look at the sides. Are any of the sides equal in length, or congruent?

no The figure is a(n)

scalene triangle .

3. Look at the angles in the figure. One angle has a measure

greater than

90°.

obtuse

The figure is a(n)

,


scalene triangle .

Example 2 Identify the figure.

1. The figure has four sides. The figure is a quadrilateral. 2. All sides are equal in length. The figure is a rhombus or a square. 3. All angles are not equal in size. The figure is a rhombus.

YOUR TURN! Identify the figure.

1. The figure has 4 The figure is a(n)

sides.

quadrilateral

.

2. Look at the sides in the figure. There is 1 pair of parallel sides. trapezoid The figure is a(n)

.

GO ON Lesson 4-3 Triangles and Quadrilaterals

75

Who is Correct? Sketch an equilateral triangle.

Aaron

Lakita

Lucinda


Guided Practice Sketch a figure with the description given. 1

obtuse, isosceles triangle

parallelogram

2

3

Identify the figure.

3 Step 1 The figure has sides and triangle The figure is a(n) .

3

angles.

Step 2 Look at the sides in the triangle. Are any of the sides no congruent? The figure is a(n) scalene triangle. Step 3 Look at the angles in the triangle.

90 °. One angle has a measure of right scalene The figure is a(n) , triangle. 76



Step by Step Practice

Identify each figure. 4

number of sides:

3

number of sides equal in length:

5

number of sides:

0

less than

measure of angles: The figure is a(n)

3

number of angles:

90°

acute, scalene triangle 4

.

4

number of angles:

number of pairs of parallel sides: measure of angles:

90

The figure is a(n)

rectangle

2

° .

6

7

The figure is a(n)

The figure is a(n)

acute, isosceles triangle




Solve. 8

SAFETY Miko saw a street sign that looked like the one at the right. Classify the shape of the sign. Understand

Solve

Look for a pattern. Guess and check. Act it out. Work backward.

Read the problem. Write what you know. Miko saw

Plan

square .

a street sign

.

Pick a strategy. One strategy is to use a diagram.

3 The street sign has sides and 3 angles. The figure is a(n) triangle . equal The sides appear to be in length. The figure is a(n) equilateral triangle .

Check

Review the definition of the figure you named.

GO ON

Lesson 4-3 Triangles and Quadrilaterals

77

9

10

FLAGS Delaware’s state flag is shown at the right. Classify rhombus the quadrilateral in the middle of the flag. Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

HOBBIES Lucille’s vegetable garden is 12 feet long on each of its four sides. Each corner forms a right angle. What type of quadrilateral does Lucille’s garden represent?

square

Can an equilateral triangle be classified as equilateral and right? Explain.

11

No; an equilateral is a triangle with three congruent sides and three congruent angles. Since the 3 angles are congruent and there are 180° in a triangle, each angle must be 180° ÷ 3 = 60°.

Sketch a figure with the description given. 12

equilateral triangle


Corbis

78

13

trapezoid



Identify each figure. 14

15

The figure is a(n)

rectangle

The figure is a(n)

obtuse, isosceles triangle .

.

16

17

The figure is a(n)

right, scalene triangle

The figure is a(n)

rhombus

.

18

19

The figure is a(n)

parallelogram

The figure is a(n)

obtuse, scalene triangle .

.

20


.

21

The figure is a(n)

trapezoid

The figure is a(n) .

acute, isosceles triangle .

Solve. 22

SHOPPING At the Purse Pagoda, the evening purses were on sale. What type of quadrilateral is the purse shown at the right?

trapezoid 23

ART Ace made a model of the Great Pyramid in Egypt. Each triangle that made his pyramid had sides of length 30 centimeters, 26 centimeters, and 30 centimeters. What type of triangle did each side make?

isosceles

GO ON Lesson 4-3 Triangles and Quadrilaterals

Comstock/Imagestate

79


trapezoid A(n) of parallel sides.

is a quadrilateral with exactly one pair

25

A(n) isosceles triangle the same length.

is a triangle with at least two sides of

26

rhombus A(n) is a parallelogram with four congruent sides. This figure may or may not have right angles.

27

A(n)

28

obtuse triangle is a triangle with one angle greater A(n) than 90° and less than 180°.

29

Writing in Math Explain how to identify a figure.

scalene triangle

is a triangle with no congruent sides.

Find the number of sides and angles of the figure. Look at the sides and angles to compare the lengths and degrees of the figure so that you can classify the figure.

30

ARCHITECTURE Presently, the famed Tower of Pisa is leaning at about a 6° angle. This makes the angle of the leaning side to the ground measure about 84°. What type of an angle is the angle of the leaning side to the ground?

acute

(Lesson 4-2, p. 63)

Find the volume of the rectangular prism.

(Lesson 2-4, p. 77)

ARCHITECTURE Tower of Pisa

31

The volume of the rectangular prism is cubic units.

80


Corbis

84


Spiral Review

Lesson

4-4 Add Angles KEY Concept Pairs of Angles

Descriptions

" ¡ #

%

%

N%#"

$

¡

¡ #

N%#$

"

This symbol means the “measure” of angle ABC.

¡ :

;

:

9

#

$

VOCABULARY complementary angles two angles that have measures with a sum of 90°

m∠ABC + m∠DBA 40° + 50° = 90°

supplementary angles two angles that have measures with a sum of 180°

9

¡

¡ ¡

complementary angles m∠ABC = 40° m∠DBA = 50°

3

9

"

5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools. 5MG2.2 Know that the sum of the angles of any triangle is 180° and the sum of the angles of any quadrilateral is 360° and use this information to solve problems. 6MG2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.

¡

¡ ¡ :

3

;


supplementary angles m∠XYZ = 115° m∠RYX = 65° m∠XYZ + m∠RYX 115° + 65° = 180° Measures of Figures

Descriptions ¡

¡

¡

¡

¡ ¡

The sum of the measures of the angles of a triangle is 180°.

4VNPGBOHMFT ¡

4VNPGBOHMFT ¡

The sum of the measures of the angles of a quadrilateral is 360°. 180° + 180° = 360° Complementary angles with a common ray form a right angle. Supplementary angles with a common ray form a straight angle.

GO ON Lesson 4-4 Add Angles

81

Example 1

YOUR TURN!

Identify the angles as complementary or supplementary. Find the missing angle. 1. The sum of the ( ¡ measures of ∠FHG ¡ Y ¡ and ∠GHJ equals 180. ' ) + The sum of the measures of complementary angles equals 180.

" # ¡ $

¡ %

m∠BCD + m∠ACB = 90 25° + ?° = 90° ?° = 65°

m∠FHG + m∠GHJ = 180 100° + ?° = 180° ?° = 80° ∠FHG and ∠GHJ are supplementary angles. m∠GHJ = 80

Example 2

3. 25° + 65° = 90° m∠ACB =

65

YOUR TURN!

Find the measure of the missing angle.

¡

)

1. Find the sum of the measures of the known angles. m∠G + m∠H = 75° + 50° = 125° 2. The sum of the measures of the angles of a triangle is 180°. Subtract the sum of the known measures from 180°. 180° - 125° = 55° The measure of the missing angle is 55°. Chapter 4 Angles and Circles

m∠Q + m∠R + m∠T

90 ° + = 245 ° =

75 ° +

5 ¡ 4

2

¡

¡

¡

3

80 °

2. The sum of the measures of the angles of a quadrilateral is 360 °. Subtract the sum of the known measures from 360 °.

360 ° - 245 ° = 115 ° The measure of the missing angle is 115 °.


¡

¡

Find the measure of the missing angle. 1. Find the sum of the measures of known angles.

'

82

1. The sum of the measures of angles equals 90 . The sum of the measures of complementary angles equals 90 .

2. Find the measure of ∠ACB.

2. Find the m∠GHJ

(

Identify the angles as complementary or supplementary. Find the missing angle.

Example 3

YOUR TURN!

Sketch supplementary angles when one angle’s measure is 45.

Sketch complementary angles when one angle’s measure is 70.

5. Find the missing angle by using the protractor’s scale. The supplementary angle is 135°.

¡

4. Find the given angle, 45°, inside the 180° angle and make a dot. Use a straightedge to draw the ray that connects the vertex and the dot at 45° from either of the two existing rays.

3. The sum of supplementary angles equals 180°. Find the angle on the protractor’s scale and make a dot. Use a straightedge to draw a second ray that connects at the vertex.

2. Draw one side of the angle and place an arrow at the end of the ray.

¡

1. Use the center point on the protractor as the common vertex of the supplementary angles.

7FSUFY

7FSUFY


¡

¡

¡

1. Use the center point on the protractor as the common vertex of the supplementary angles. 2. Draw one side of the angle and place an arrow at the end of the ray. 3. The sum of complementary angles equals 90° . Find the angle on the protractor’s scale and make a dot. Use a straightedge to draw a second ray that connects at the vertex. 4. Find the given angle, 70°, inside the 90° angle and make a dot. Use a straightedge to draw the ray that connects the vertex and the dot at 70° from either of the two existing rays. 5. Find the missing angle by using the protractor’s scale. The complementary angle is 20°.

Who is Correct? ∠MNO and ∠PNQ are supplementary angles. The measure of ∠MNO is 67°. What is the measure of ∠PNQ?

Diego

Patrick

90° - 67° = 23°

Malia

180° - 67° = 123°

180° - 67° = 113°



83

Guided Practice Identify the angles as complementary or supplementary. Find the missing angle. 1

complementary . The angles are The measure of the missing angle is 75° .

2

supplementary . The angles are The measure of the missing angle is 105° . 0

,

¡

¡ /

¡

¡ ¡

.

2

1

4


Find the measure of the missing angle. Step 1

Find the sum of the measures of known angles. m∠D + m∠E =

49 ° +

63 ° = 112 °

Step 2 The sum of the measures of the angles of a triangle is 180 °. Subtract the sum of measures of the known angles from 180 °.

180 ° - 112 ° =

" ¡ ¡ % ¡

¡ $

84 ° + 143 ° + 103 ° = 330 ° 360 ° - 330 ° = 30 ° The measure of the missing angle is 30 °. m∠A + m∠C + m∠D =

84


&

¡

¡

'


68 °.

Find the measure of each missing angle.

#

¡

68 °

The measure of the missing angle is

4

%

Find the measure of each missing angle. 5

: ¡

¡

¡

9

;

58 ° + = 155 ° 180 ° - 155 ° = 25 °

97 °

m∠Y + m∠Z =

The measure of the missing angle is 6

25 °. 7

-

) ¡

¡

*

.

¡

¡

¡

, +

/

The measure of the missing angle is 90 °.


Sketch each type of angle given. Sketch supplementary angles when one angle’s measure is 85°.

9

Sketch complementary angles when. one angle’s measure is 41°.

8

3

¡ ¡

4

5

3

¡ ¡

8


¡ ¡

7


85



Solve. 10

ART Samir drew his version of a pine tree. It was the shape of a triangle. It had a 36° angle and a 70° angle. What was the measure of the third angle of Samir’s pine tree? Understand

Look for a pattern. Guess and check. Act it out. Solve a simpler problem.

Read the problem. Write what you know. The measure of one angle of Samir’s pine tree was 36 °. The measure of the other angle was 70 °.

Plan

Pick a strategy. One strategy is to draw a diagram.

Solve

Find the sum of the measures of the known angles.

36 ° +

70 ° = 106 °

36°

Subtract this sum from 180°. 180° - 106 ° =

The measure of the third angle is

36° + 70° +

74 ° = 180 °

AGRICULTURE A cow pasture is shaped like a parallelogram. Two of the angles measure 68° and 112°. What are the measures of the 68° and 112° third and fourth angles? Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

SPORTS Hughes Hardware engraves trophies. The trophies have sides in the shape of isosceles triangles. The base angles of each side measure 73°. How many degrees is the top angle?

34° 86

74 °.



12

?

Add the three angle measures. The sum should equal 180°.

Check

11

70°

74 °

Explain how you can find the sum of the angle measures in a pentagon.

13

Divide the pentagon into triangles. Since the sum of the angles of each triangle equals 180°, the sum of the angle measures of a pentagon must equal 3(180), or 540°.

Skills, Concepts, and Problem Solving Identify the angles as complementary or supplementary. Find the missing angle. 14

complementary . supplementary . 15 The angles are The angles are 65° The measure of the missing angle is . The measure of the missing angle is 95° . ,

. ¡

-

¡

¡

¡

/



16

17

1 ¡

¡ ¡

¡

5

6

¡

2

The measure of the missing angle is 113 °. 18 ,

+

¡

3

The measure of the missing angle is 105 °. 19

-

4 ¡

:

¡ ¡

¡

¡

¡

9

¡

¡

;

.



GO ON

Lesson 4-4 Add Angles

87

Sketch each type of angle given. Sketch complementary angles when one angle’s measure is 58°.

21

Sketch supplementary angles when one angle’s measure is 52°.

'

,

)

(

+

20

-

Solve. CROSS-COUNTRY A cross-country team ran on a path near the school that was in the shape of a quadrilateral. Three angles of the area measured 42°, 85°, and 102°. How many degrees was the fourth angle? 131°

23

SCULPTURE The geometric sculpture that sits at the entrance of Ugo Park has a triangle for its base. It has angles that measure 38° and 104°. What is the measure of the third angle? 38°

24

PETS The play area at Pooch Paradise is in the shape of a quadrilateral. Three angles of the area measure 25°, 112°, and 82°. How many degrees is the fourth angle? 141°

25

GARDENING The butterfly garden at Sweet Valley Academy is placed in the corner of the courtyard. It is in the shape of a triangle. The butterfly garden has angles that measure 62° and 47°. What is the measure of the third angle? 71°


Complementary

angles are two angles that have measures

with a sum of 90°. 27

Supplementary

angles are two angles that have measures

with a sum of 180°.

88



22

28

Writing in Math Explain how to find the missing angle of a quadrilateral when the measures of three angles are given.

Sample answer: Find the sum of the known angle measures. The sum of the measures of the angles of a quadrilateral is 360°. Subtract the sum of the known angle measures from 360°.

Spiral Review 29

LETTERS Which letter in the name KEN forms perpendicular lines? E (Lesson 4-1, p. 54)


(Lesson 3-5, p. 37)

30

31



Find the area of each rectangle.


(Lesson 3-1, p. 4)

32

33 GU DN

GU DN

The area of the rectangle is 108 square centimeters or 108 cm2.

The area of the square is 49 ft2. square feet or

49

Lesson 4-4 Add Angles

89

Chapter

Progress Check 2

4


Draw a figure with the description given. 5MG2.1 1

acute, scalene triangle See TWE margin.

2

trapezoid See TWE margin.

4

Draw complementary angles.

Draw each type of angles given. 5MG2.1 Draw supplementary angles.

$

%

#

"

3

'

&

Classify each figure. 5MG2.1 5

6

The figure is a right, isosceles triangle .

rhombus

The figure is a

7

8

(

+

-

¡

¡

¡

, )

¡

¡

¡


Solve. 5MG2.1, 5MG2.2 REPAIRS A school bus was placed on a lift so that the repair workers could see under the bus. What type of quadrilateral does the side of the lift represent?

parallelogram 90

.

*


9

¡



Find the measure of each missing angle. 5MG2.2, 6MG2.2

.

Lesson

4-5 Congruent Figures 7MG3.4 Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationships between the sides and angles of the two figures.

KEY Concept /

2 ¡

¡

VOCABULARY ¡

¡

.

¡

0

¡

corresponding sides −−− −− MN PQ −−− −− MO PR −−− −− NO QR

congruent figures figures having the same size and the same shape

3

1

corresponding angles

corresponding angles angles of congruent figures that have the same measure

m∠M m∠P m∠N m∠Q m∠O m∠R

MNO and PRQ are congruent because corresponding sides and corresponding angles are congruent.

corresponding sides sides of congruent figures that have the same length

5IFTZNCPMɁNFBOT iJTDPOHSVFOUUPw

The number of hash marks on a line segment indicates which sides are congruent.


Example 1

YOUR TURN!

STU and FGH are congruent. Identify corresponding sides and corresponding angles. )

5

Quadrilateral ABCD and quadrilateral WXYZ are congruent. Identify corresponding sides and corresponding angles. "

¡ ¡ ( 4

¡

¡

%

¡

¡

¡

:

#

¡ ¡

¡ $

¡ 6

¡

'

1. Find the corresponding sides. −− −− ST corresponds to FG. −− −− SU corresponds to FH. −−− −− TU corresponds to GH. 2. Find the corresponding angles. ∠S corresponds to ∠F. ∠T corresponds to ∠G. ∠U corresponds to ∠H.

9

; ¡ ¡ 8

1. Find the corresponding sides. −− −− −− WX AB corresponds to corresponds to −− ; BC −−− −−− CD corresponds to YZ ; AD corresponds to

−− XY . −− WZ .

2. Find the corresponding angles.

∠A corresponds to ∠W ; ∠B corresponds to ∠X . ∠C corresponds to ∠Y ; ∠D corresponds to ∠Z .

Lesson 4-5 Congruent Figures

91

Example 2

YOUR TURN! Are the quadrilaterals congruent?

Are the triangles congruent?

0 4

7

¡

¡ 3

¡

3

¡

5

8

¡

¡

4

¡

¡

¡

¡

1

6 2

1. Find the corresponding sides. −− −−− RS WV

¡

¡

¡

5

¡

/

.

1. Find the corresponding sides. −− −− −−− −− MN QR −− RS NO −− −− −− ST OP QT MP

2. Find the corresponding angles. m∠S m∠V

2. Find the corresponding angles.

RST and WVU are not congruent because these triangles do not have 3 corresponding sides and 3 corresponding angles.

m∠Q m∠ M m∠R m∠ N m∠S m∠ O m∠T m∠ P Quadrilateral QRST and quadrilateral congruent MNOP are because corresponding sides and corresponding congruent angles are .


Who is Correct? Mrs. Johnson asked her students to draw two congruent triangles.

Andrea

Shannon

¡

+

¡ 5

¡

/

¡

+

, 3

¡

¡

¡

-

¡

¡

¡


¡

¡

¡

¡

6 9

¡

,

5 7

-

Circle correct answer(s). Cross out incorrect answer(s). 92

0

4

¡

David

.

%

¡

¡

8

Guided Practice Sketch two congruent figures. 1

triangles 0

7

¡

/

¡

quadrilaterals

2

3

¡

¡

¡

¡ 4

¡

¡

1 8

.

¡

¡

0

6 ¡ 2

¡ 5

¡ /

¡ 1

Step by Step Practice Are the triangles congruent? -

3 *

¡

¡

)

¡

¡

¡ .

+

¡


/

Step 1

Find the corresponding sides.

−− −− −− −−− HI and LM are corresponding sides; LM HI . −− −− − − IJ and MN are corresponding sides; IJ MN . −− −− −− −− JH and NL are corresponding sides; JH NL . Step 2

Find the corresponding angles. ∠H and ∠ L

are corresponding angles; m∠H m∠ L .

∠I and ∠ M are corresponding angles; m∠I m∠ M . ∠J and ∠ N

are corresponding angles; m∠J m∠ N .

congruent HIJ and LMN are because corresponding sides and corresponding congruent angles are . GO ON Lesson 4-5 Congruent Figures

93

%

4

¡ . ¡ -

¡

¡

/

$

¡ ¡ #

Find the corresponding sides. −− −−− LM BC −− −−− MN CD −− −− NL DB

Find the corresponding angles. ∠L ∠B ∠M ∠C ∠N ∠D

congruent LMN and BCD are because corresponding congruent sides and corresponding angles are . Are the figures congruent? Quadrilateral ABCD and quadrilateral not congruent because VWXY are corresponding sides and corresponding not congruent . angles are

congruent EFG and JKL are because corresponding sides and corresponding angles are congruent .


¡

¡ ¡

¡

¡ %

¡ 7

8

'

, ¡

&

94

:

¡

¡ #

"

6

9

$

¡

¡

¡

( -

¡

¡

+


5


Problem-Solving Strategies Draw a diagram. Look for a pattern. Guess and check. ✓ Act it out. Solve a simpler problem.

Solve. OFFICE Lester saw two stacks of plain white paper by the copy machine. He decided that it did not matter which stack of paper he used because both have four right angles. Below is a picture of each sheet of paper Lester could use. Are the sheets of paper congruent?

7

Understand

Read the problem. Write what you know. Each sheet of paper has four right angles. Each sheet of paper is a rectangle .

Plan

Pick a strategy. One strategy is to act it out.

Solve

Measure the side lengths of each sheet of paper. Do the corresponding sides of each piece of paper no have the same length? The sheets of paper

Trace the sheets of paper. Cut them and place them at top of each other. Do the sheets of paper have the same shape and size? no Can they be rotated to have the same shape and size? no

Check


are not congruent.

. 8

9

TRIANGLES Are triangles ABC and XYZ congruent? yes Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

$

;

¡

"

¡

¡

#

:

¡

9

FLAGS An image of the United States flag is shown. Are the yes stars on the flag congruent? Are all triangles that have angle measures 30°, 60°, and 90° congruent? Explain.

10

No; the side lengths of the triangles could be different.

GO ON Lesson 4-5 Congruent Figures

Hespenheide Design

95

Skills, Concepts, and Problem Solving Sketch two congruent figures. 11

quadrilaterals "

¡

12 3

#

¡

¡ $

"

¡

2

¡

¡

(

¡

#

5

¡

¡

%

triangles

¡

¡

¡

$

'

¡

¡

)

4

Are the figures congruent? +

¡ & ¡ %

13

¡

'

* ¡¡ )

Find the corresponding sides.

−− HI − IJ −− JH

Find the corresponding angles.

∠D ∠H ∠E

∠I

∠F

∠J

congruent DEF and HIJ are sides and corresponding angles are

96


because corresponding .

congruent


−− DE −− EF −− FD

14

¡

Are the figures congruent? 15

Quadrilateral PQRS and quadrilateral FGHI congruent are because corresponding sides and corresponding congruent angles are .

*

)

¡

2 ¡

3

Solve. 16

¡

¡ '

¡

POLYGONS There are some polygons that are congruent to each other. Which of the following polygons are congruent to pentagon ABCDE?

¡

¡

1

4

¡ (

$

,

' ¡ ¡ ¡

%

¡

¡

+ ¡

#

¡ ¡ "

¡ (

¡ ¡

*

&

)

:

¡

¡ 6 ¡ ¡ 7

0 ¡ /

¡ ¡

¡

.

¡

9

8

pentagon UYXWV 17

BADGES Mrs. O’Shay is designing new badges for the safety patrol officers. She has to decide between the two designs pictured at the right. Are the figures congruent? Explain.

No; the figures


are not the same shape. Vocabulary Write the vocabulary word that completes each sentence. 18

Congruent figures have the same size and the same shape. angles

19

Congruent same size.

20

Congruent length.

21

Writing in Math Explain how to determine if two triangles are congruent.

sides

are angles that have the are sides that have the same

Find corresponding sides and angles. If the corresponding sides and angles are congruent, then the two triangles are congruent. GO ON Lesson 4-5 Congruent Figures

97

Spiral Review 22

TILE Each shower tile in Danica’s bathroom is shaped like a rhombus. One angle of each tile measures 115°. What are the measures of the other three angles? (Lesson 4-4, p. 81)

115°, 65°, and 65°

Measure and classify each angle. 23

(Lesson 4-2, p. 63)

30 ∠JKL measures °. ∠KL is a(n) acute angle because its angle measure is greater than 0° but less than 90°. +

¡

,

24

-

∠FGH measures

90

°. ∠FGH is a(n)

right

angle.

'

(

¡ )

25

The area of the square is square units.

98


(Lesson 2-3, p. 71)

64

26

The area of the rectangle is square units.

42


Find the area of each figure.

Lesson

4-6 Pythagorean Theorem KEY Concept The Pythagorean Theorem 2 2 2 states that a +b =c

.

D

5MG2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools. 7MG3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, IZQPUFOVTF empirically verify the Pythagorean theorem by direct measurement.

B

VOCABULARY MFHT

2

2

a +b = c 2 2 3 + 4 = 52

2

(3×3) + (4×4) = (5×5) 9 + 16 = 25 25 = 25 Copyright © by The McGraw-Hill Companies, Inc.

legs of a triangle two sides that form the right angle in a right triangle

To prove the Pythagorean Theorem, count the number of squares beside each leg of the triangle to the left. The total is equal to the number of squares beside the hypotenuse. You can also use the formula.

C

hypotenuse side opposite the right angle in a right triangle Pythagorean Theorem the sum of the squares of the lengths of the legs in a right triangle is equal to the square of length of the hypotenuse

Use the formula. Substitute the measure of each side.

square root one of the two equal factors of a number

Simplify the exponents. Multiply. Add.

square number the product of a number multiplied by itself

You can use the Pythagorean Theorem to find the length of the hypotenuse given the lengths of the legs, or the length of one leg given the lengths of the hypotenuse and the other leg. A Note on Square Numbers and Square Roots: A square number, such as 42, indicates that 4 should be multiplied by itself 2 times. 2

4 = 4 × 4 = 16 To find a square root , such as √ 16 , think about what number times itself equals 16. √ 16 = √ 4×4 =4

(-4) × (-4) is another square root of 16. Every positive number has two different square roots. The symbol √ is used to show the positive square root of a positive number; for example, √ 16 = 4.

GO ON Lesson 4-6 Pythagorean Theorem

99

Example 1 Find the length of the hypotenuse of the right triangle. 1. From the figure, you know a = 6 and b = 8.

D

2. Substitute the values of a and b in the Pythagorean Theorem and solve for c. 2

2

2

Use the formula.

2

2

6 +8 =c

2

Substitute the measure of each side.

36 + 64 = c

2

Simplify the exponents.

2

Add 36 and 64.

a +b =c

100 = c √ 100 = √ c2 10 = c

Take the square root of both sides. Simplify the square root.

The length of the hypotenuse is 10 units. YOUR TURN! Find the length of the hypotenuse of the right triangle.

1. From the figure, you know a=

9

and b =

D

12 .

2. Substitute the values of a and b in the Pythagorean Theorem and solve for c. 2

2

a +b =c 2

81

+

12

+ 144

2

=c

2

=c

2

225 = c 2 225 = √ c2 15

=c

The length of the hypotenuse is

15

units.

Numbers like 4, 9, 25, 49, and 100 are called perfect squares because their square roots are whole numbers.

100



9

2

Example 2 Find the length of the leg of the right triangle. 1. From the figure, you know b = 60 and c = 75. 2. Substitute the values of b and c in the Pythagorean Theorem and solve for a. a2 + b2 = c2 2

2

Use the formula. 2

a + 60 = 75

B

Substitute the measure of each side.

a2 + 3,600 = 5,625

Simplify the exponents.

a2 = 5,625 - 3,600

Subtract 3,600 from 5,625.

2

a = 2,025 √a2 = √ 2,025

Simplify. Take the square root of both sides.

a = 45

Simplify the square root.

The length of the leg is 45 units. YOUR TURN! Find the length of the leg of the right triangle. 1. From the figure, you know 12 20 a= and c = .

C

2. Substitute the values of a and c in the Pythagorean Theorem and solve for b.


a2 + b2 = c2

122 144

+ b2 = + b2 = b2 = b2 = √b2 = b=

202 400 400-144 256 √ 256 16

The length of the leg is

16

units.

Whenever you have a right triangle, then you know its sides follow the Pythagorean Theorem, a2 + b2 = c2. You can also do this in reverse. If you have a triangle and you know that the sum of the squares of the two smaller sides equals the square of the larger side (its sides follow the Pythagorean Theorem), then the triangle must be a right triangle.

GO ON

Lesson 4-6 Pythagorean Theorem

101

Example 3

YOUR TURN! Determine if a triangle with the following lengths is a right triangle.

Determine if a triangle with the following lengths is a right triangle. 13

5

15

8 12

12

Determine which side is the largest. 13

Determine which side is the largest.

Substitute the values into a2 + b2 = c2, ensuring that the largest number is c, and solve. a2 + b2 = c2 52 + 122 = 132 25 + 144 = 169 169 = 169

Use the formula. Substitute the measure of each side. Simplify the exponents. Add 25 and 144

Substitute the values into a2 + b2 = c2, ensuring that the largest number is c, and solve.

Since this is a true statement, the triangle is a right triangle.

a2 + b2 = c2

82 64

122 = 152 + 144 = 225 208 ≠ 225 +

Since this is not a true statement, the triangle is not a right triangle.

Mr. Fernandez told his students to draw and label the sides of a right triangle. Which student’s triangle is correct?

Rhonda

13 + 14 = 27

2 2 1002 - 60 = 80


102


Sam

Pete

2 2 602 + 80 = 100


Who is Correct?

Guided Practice Use the measurements given to draw a right triangle or indicate if a right triangle cannot be drawn with the measurements. 1

4 cm, 3 cm, and 5 cm

2

A right triangle cannot be drawn with these measurements.

DN

DN

15 in., 20 in., and 30 in.

DN


Find the length of the hypotenuse of the right triangle.

D

Step 1 Find the lengths of the legs, a and b. a=

8

15

and b =

Step 2 Substitute the values of a and b in the Pythagorean Theorem and solve for c. 2

2

a +b =c

8

2


64

2

15 2 = c 2 2 + 225 = c 289 = c 2 c2 √ 289 = √ 17 = c

+


17

units.

Find the length of the leg or hypotenuse of each right triangle. 2

2

a +b =c

4

2

2 a + 12 = 13 2 a + 144 = 169 2 a = 25 √ a 2 = √25 a= 5 2

B

2


5

units. GO ON Lesson 4-6 Pythagorean Theorem

103

5

B


7

units.

6 D


7

50

units.

B


9

units.

8

yes; 182 + 802 = 822

9

no; 32 + 52 ≠ 49

104



Determine if each triangle is a right triangle.



Solve. 10

Understand

Read the problem. Write what you know. The ladder touches the wall at a height of feet. The ladder is wall.

9

12

feet away from the base of the

Plan

Pick a strategy. One strategy is to draw a diagram. Then, use the Pythagorean Theorem to find the length of the ladder.

Solve

Substitute 9 for a and 12 for b in the Pythagorean Theorem and solve for c. 2

2

a +b =c


Look for a pattern. Act it out. Solve a simpler problem. Work backward.

HOME IMPROVEMENT Jerry’s ladder is resting against a wall. The top of the ladder touches the wall at a height of 12 feet. The bottom of his ladder is 9 feet away from the base of the wall. How long is Jerry’s ladder? 15 feet

D

2

9 2 + 12 2 = c 2 81 + 144 = c 2 225 = c 2 √ 225 = √ c2 15 = c The height of the ladder is

GU

15

feet.

Use a calculator to check your answer.

Check

11

GU

CONSTRUCTION A staircase to the attic in Cassandra’s house has a length of 10 feet. The top of the stairs meets the wall at a height of 8 feet. How far away is the bottom of the staircase from the wall? 6 feet Check off each step.

✔

Understand

✔

Plan

✔

Solve

✔

Check

GO ON Lesson 4-6 Pythagorean Theorem

105

12

ADVERTISING Splash Village advertised a water slide that is 20 yards long and 12 yards from the base of the steps to the end of the slide. What is the height of the steps that reach the top of the slide? 16 yards Can line segments with lengths 30 inches, 30 inches, and 50 inches form a right triangle? Explain.

13

No; substitute 30 for a, 30 for b, and 50 for c in the Pythagorean Theorem: 2

2

2

30 + 30 50 ; 900 + 900 2,500; 1,800 ≠ 2,500.

Skills, Concepts, and Problem Solving Use the measurements given to draw a right triangle or indicate if a right triangle cannot be drawn with the measurements. 14

5 cm, 10 cm, and 15 cm

15


6 cm, 8 cm, and 10 cm DN DN

DN

Find the length of the leg or hypotenuse of each right triangle. 16

17

D


12

units.

The length of the hypotenuse is 13 units.

Determine if each triangle is a right triangle. 18

19

20

16

12

no; 72 + 82 ≠ 122

106


yes; 122 + 162 = 202


B

Solve. 20

TRAVEL Reggie left his house and drove 20 miles east and then 15 miles south. If Reggie follows a straight line to his house, how far is Reggie from his house?

NJFBTU

NJTPVUI

25 miles 21

DISTANCE A lighthouse on a cliff near Gina’s house is 12 meters tall. Gina stood 9 meters away from the base of the lighthouse. How far away was Gina from the top of the lighthouse?

15 meters Vocabulary Check Write the vocabulary word that completes each sentence. 22

hypotenuse A(n) right triangle.

is the side opposite the right angle in a

23

square number A(n) by itself.

is the product of a number multiplied

24

25

Pythagorean Theorem The states that the sum of the squares of the lengths of the legs in a right triangle is equal to the square of length of the hypotenuse. Writing in Math Explain how to find the length of the hypotenuse of a right triangle when the lengths of the legs are known. 2


Substitute the values of a and b in the Pythagorean Theorem (a 2 + b = c 2) and solve for c.

Spiral Review 26

POLYGONS Which two of the following polygons are congruent? (Lesson 4-5, p. 91) 120°

60° 60°

60°

120° 150°

60°

60°

120°

30°

30°

150°

60°

120° 30°

30°

60°

120°

The second and fifth polygons are congruent.

Lesson 4-6 Pythagorean Theorem

107

Chapter

Progress Check 3

4


Draw two congruent figures. 5MG2.1, 5MG2.2 1

triangles

2

See TWE margin.

quadrilaterals

See TWE margin.

Use the measurements given to draw a right triangle or indicate if a right triangle cannot be drawn with the measurements. 5MG2.1, 7MG3.3 3

1.3 cm, 8.4 cm, and 8.5 cm

4

12 mm, 13 mm, and 20 mm


DN DN DN

5

Are the figures congruent? 7MG3.4 -

6

2 ¡

¡

¡

1

¡

.

"

¡ ¡

Find the length of the leg of the right triangle. 7MG3.3

B

;


10

units.

Solve. 7MG3.3 7

FLAGS A flagpole stands in front of a city building. The base of the flagpole is 8 yards away from a spotlight that shines on the flag. The distance from the spotlight to the top of the flagpole is 10 6 yards yards. How tall is the flagpole? EJBHPOBM

8

RECTANGLES What is the length of the diagonal 13 of rectangle MNOP?

. 1

108


/ D

0


PLM and QAZ are not congruent because corresponding sides and corresponding angles are not congruent .

Lesson

4-7 Circles

6MG1.2 Know common estimates of pi (3.14, 22 ) and 7 use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements.

_

KEY Concept A circle is the set of all points in a plane that are the same distance from a point called the center. DFOUFS SBEJVT S

circle a closed figure in which all points are the same distance from a fixed point called the center of the circle

E EJBNFUFS

DJ

SD V

Formulas

NGFSFO

DF

Descriptions The length of the diameter is twice the length of the radius of a circle.

d = 2r diameter

S

radius

E S

circumference


C = πd

C = 2πr

or

diameter

radius

A = πr2 Area

radius

VOCABULARY

The ratio of the circumference of every circle to its diameter is always the same number: π C =π d

_

The area of any circle is always equal to π times the radius squared.

Pi (π) is an irrational number. It is not possible to write the exact value of π. To calculate using π, either use a calculator with a π button, or substitute one of the approximate values 22 . of π, 3.14 or ___ 7

chord a line segment with both endpoints on a circle diameter a chord that passes through the center of a circle radius a line segment that connects the center of a circle to a point on the circle circumference the distance around a circle pi (π) the ratio of the circumference of a circle to the diameter of the same circle The value of π is 22 approximately 3.14 or ___. 7

When you make a substitution for π, you have to change the = symbol to ≈ symbol to show that you are making an estimate. GO ON Lesson 4-7 Circles

109

Example 1

YOUR TURN! What is the circumference of the circle?

What is the circumference of the circle? 1. The diameter is 6 centimeters; d = 6.

GU DN

2. Substitute 6 for d and 3.14 for π in the circumference formula. C = πd C ≈ 3.14 × 6 C ≈ 18.84 The circumference of the circle is about 18.84 centimeters.

1. Find the length of the radius. The radius is so r = 8 .

8

feet,

2. Substitute 8 for r and 3.14 for π in the circumference formula. C = 2πr C ≈ 2 × 3.14 × C ≈ 50.24

8

The circumference of the circle is about

50.24 feet.

Example 2

YOUR TURN! What is the area of the circle?

What is the area of the circle? ZE

2. Substitute 10 for r and 3.14 for π in the area formula. A = πr2

1. Find the length of the radius.

The diameter is 14 inches. r = 14 ÷ 2 = 7 . 22 for π in 2. Substitute 7 for r and ___ 7 the area formula.

A ≈ 3.14 × 102

A = πr2

A ≈ 3.14 × 100

22 × A ≈ ___ 7 22 × A ≈ ___ 7

2200 A ≈ _____ or 314 7 The area of the circle is about 314 square yards.

7

2

49

A ≈ 154 The area of the circle is about

154 square inches.

110


JO


1. The radius is 10 yards; r = 10.

Who is Correct? What is the circumference of a circle with a radius of 5 yards?

Tavio

Yang

Amanda

C = 2πr C ≈ 2 × 5 × 3.14

C = πr C ≈ 5 × 3.14 The circumference is about 15.7 yd.

C = πd 10 × 3.14 ≈ C

The circumference is about 31.4 yd.

The circumference is about 31.4 yd.


Guided Practice Identify the length of the radius and the diameter of each circle. 1

2 N


GU

radius: 2 ft diameter: 4 ft

3

radius: 14 m diameter: 28 m

4 N DN

radius: 32 cm diameter: 64 cm

33 in.

radius: 16.5 in. diameter: 33 in.

GO ON Lesson 4-7 Circles

111

Step by Step Practice Find the circumference and area of the circle. 5

DN

Step 1 Find the length of the diameter and radius. The diameter is

9

Step 2 For circumference, substitute formula. C = πd C ≈ 3.14 × C ≈ 28.26

r=

centimeters.

9

9

÷ 2 = 4.5 .

for d and 3.14 for π in the circumference

9 4.5

Step 3 For area, substitute A = πr2 4.5 A ≈ 3.14 × A ≈ 3.14 × 20.25 A ≈ 63.585

for r and 3.14 for π in the area formula.

2

Find the circumference and area of each circle. Use 3.14 for π. 6

26 d= C = πd C ≈ 3.14 × C ≈ 81.64

in.

26

13 r= in. 2 A = πr 13 A ≈ 3.14 × 169 A ≈ 3.14 × A ≈ 530.66

2

JO

The circumference of the circle is about 81.64 inches, and the area of the circle is about 530.66 square inches. 7

The circumference of the circle is about 21.98 meters, and the area of the circle is about 38.465 square meters. N

112



The circumference of the circle is about 28.26 centimeters, and the area of the circle is about 63.585 square centimeters.

Find the circumference and area of each circle. Use 8

14 d= C = πd 22 × C ≈ ___ 7 C≈

in.

44

44

7 r= in. 2 A = πr 22 × 7 A ≈ ___ 7 22 × 49 A ≈ ___ 7 154 A≈

22 _ for π. 7

ZE 2

44 yards, and the The circumference of the circle is about 154 area of the circle is about square yards. 9

_

88 _

The circumference of the circle is about 7 88 area of the circle is about square feet.

feet, and the GU

7

Step by Step Problem-Solving Practice Solve.


10

CARS Alana’s tires each have a diameter of 35 inches. What is the circumference of each tire? Understand

Read the problem. Write what you know. Alana’s tires have a diameter of 35 inches.

Plan

Pick a strategy. One strategy is to use a formula. Use the formula for the circumference of a circle.

Solve

Problem-Solving Strategies Draw a diagram. ✓ Use a formula. Guess and check. Act it out. Solve a simpler problem.

Substitute 35 for d and 3.14 for π in the circumference formula. C = πd C ≈ 3.14 × C ≈ 109.9

35

The circumference of each tire is about 109.9 inches. Check

Estimate the circumference by substituting 3 for π and 35 for d. C = πd C ≈ 3 × 35 C ≈ 105 The circumference of each tire is close to the estimate, so the answer is reasonable.

GO ON

Lesson 4-7 Circles

113

11

SPORTS The skating rink has a radius of 4 yards. What is the circumference 25.12 of the skating rink? Use 3.14 for π. Check off each step.

✔ ✔ ✔ ✔ 12

Understand Plan Solve Check

HOBBIES J.C. collects clocks. The face of his largest clock has a radius of 23 inches. What is the area of J.C.’s largest clock? Use 3.14 for π.

1,661.06 in2

13

Using an appropriate measuring tool and the centimeter grid shown, find the actual circumference and area of the circle at the right. Next, find the circumference and area using the formulas C = 2πr and A = πr2. Write down both sets of numbers. How do the two sets of numbers compare? Explain the differences.

DN

C = 2πr So, the circumference of this circle is 2 · π · 2, or 4π. The area of the circle is πr 2 or π22, which is also equal to 4π.

Skills, Concepts, and Problem-Solving Identify the length of the radius and diameter of each circle. 15 LN JO

radius: 25 in. diameter: 50 in. 16

radius: 90 km diameter: 180 km 17 DN

GU

radius: 3.5 ft diameter: 7 ft 114


radius: 12.5 cm diameter: 25 cm


14

Find the circumference and area of the circle. Use 3.14 for π. 18

47.1 yards, and the The circumference of the circle is about area of the circle is about 176.625 square yards.

22 _ for π.

_

Find the circumference and area of the circle. Use 19

ZE

132 7 The circumference of the circle is about feet, and the 7 198 square feet. area of the circle is about

_

GU

7

Solve. 20

ASTRONOMY The diameter of an eye-piece lens for a telescope is 2.4 centimeters. What is the circumference of the lens? Use 3.14 for π.

7.536 cm

21

HOBBIES Dane’s grandmother used 5 circular canvases last week while painting a collage. Each canvas had a diameter of 14 inches. What was the area of each canvas? Use 3.14 for π.

153.86 in2

Vocabulary Check Write the vocabulary word that completes each sentence. Copyright © by The McGraw-Hill Companies, Inc.

22 23

Circumference

is the distance around a circle.

radius A(n) is a line segment that connects the center of a circle to a point on the circle. Pi (π)

24

is the ratio of the circumference of a circle to the diameter of the same circle. Its value is approximately 3.14 22 . or ___ 7

25

Writing in Math Explain how to find the area of a circle.

Find the length of the radius. Substitute the length of the radius for r and 3.14 or 22 for π in the area formula. 7

_

GO ON Lesson 4-7 Circles

115

Spiral Review 26

FOOD The delivery ramp that leads to the back door of Rey’s Wholesale Foods is 5 feet long. There is a height of 3 feet from the top of the ramp to the base of the wall. How far is it from the bottom of the ramp to the base of the wall? (Lesson 4-6, p. 99) 4 feet

Find the measure of the missing angle. 27

(Lesson 4-4, p. 81)


28


#

9

¡

"

¡

¡

¡

¡ $ 8

¡

:

¡ ;

Draw a net for a rectangular prism with the dimensions given. (Lesson 3-4, p. 27)

29

7×3×6

"

%

116


#

$


!

Lesson

4-8 Volume of Triangular Prisms

6MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.

and Cylinders KEY Concept To find the volume of a triangular prism, you must first know the area of its base. The base of a triangular prism is a triangle. 1 bh. The formula for the area of a triangle is A = __ 2 The volume of a triangular prism is the product of the area of its base, B, I and its height, h. C

V is the volume of the solid figure.

B is the area of the base.

volume the number of cubic units needed to fill a threedimensional figure or solid figure (Lesson 2-4, p. 77)

I

/PUJDFUIBUUIFSFBSFUXPI T JOUIFEJBHSBN5IFIFJHIUPG UIFCBTFUSJBOHMFJTEJGGFSFOUUIBO UIFIFJHIUPGUIFUSJBOHVMBSQSJTN

V=B×h

VOCABULARY

h is the height of the base of the triangular prism.

To find the volume of a cylinder, you must first know the area of its base. The base of a cylinder is a circle. The formula for 2 the area of a circle is A = π r .

triangular prism a prism whose bases are triangular with parallelograms for sides cylinder a three-dimensional figure having two parallel congruent circular bases and a curved surface connecting the two bases


The volume of a cylinder is the product of the area of its base, B, and its height, h. V is the volume of the solid figure.

S

r is the radius.

V = B × h or V = π r 2h B is the area of the base.

I

h is the height.

As with circles, when you are calculating the volume of a cylinder and you substitute approximate values in for π, change from using an = symbol to ≈ symbol.

GO ON Lesson 4-8 Volume of Triangular Prisms and Cylinders

117

Example 1

YOUR TURN!

What is the volume of the triangular prism?

What is the volume of the triangular prism?

DN

GU

DN

1. The length of the base of the triangle is 5 centimeters, and the height of the triangle is 6 centimeters. 1 bh A = __ Use the area of a triangle formula 2 to find the area of the base. 1 × 5 × 6 Substitute the base and height of A = __ 2 the triangle. A = 15 The area of the base is 15 centimeters. 2. Find the volume by using the area of the base. V=B×h Use the formula for the volume of a triangular prism. V = 15 × 8 Substitute 8 for the height of the prism. V = 120 Multiply.

1. The length of the base of the triangle is 9 feet, and the height of the triangle is 4 feet. 1 bh A = __ 2 1× 9 × 4 A = __ 2 A = 18 2. Find the volume by using the area of the base. V=B×h V=

18

×

17

V = 306 The volume of the triangular prism is 306 cubic feet.

Example 2 What is the volume of the cylinder? 1. The radius of the circle is 4 inches. Find the area of the base. Use 3.14 for π. 2 A = πr Use the formula for the area of a circle. 2 A ≈ 3.14 × 4 Substitute the values for pi and the radius. A ≈ 50.24 Multiply. The area of the base is about 50.24 inches. 2. Find the volume by using the area of the base. V=B×h Use the formula for the volume of a cylinder. V ≈ 50.24 × 10 Substitute the values of the area of the circle and the height of the cylinder. V ≈ 502.4 Multiply. The volume of the cylinder is about 502.4 cubic inches. 118


JO

JO


The volume of the triangular prism is 120 cubic centimeters.

GU

GU

DN

YOUR TURN! What is the volume of the cylinder? 1. The diameter of the circle is 12 meters, so the radius is 12 ÷ 2 = 6 . Find the area of the base. Use 3.14 for π.

N N

2

A = πr A ≈ 3.14 × A ≈ 113.04

6

2

2. Find the volume by using the area of the base. V=B×h V ≈ 113.04 V ≈ 904.32

×

8

The volume of the cylinder is about

904.32 cubic meters.

Example 3 What is the volume of the triangular prism?

DN C


1. Use the Pythagorean Theorem to find the leg of the base of the prism. This leg is the height of the triangular base. a2 + b2 = c2 82 + b2 = 172 64 + b2 = 289 b2 = 225 b = 15

Write the formula. Substitute the measure of the sides. Simplify the exponents. Subtract 64 from 289. Find the square root of 225.

2. Next, find the area of the base. 1 bh A = __ Use the formula for the area of a triangle. 2 1 × 8 × 15 Substitute 8 for the base and 15 for the height. A = __ 2 A = 60

DN DN

DN DN DN

Multiply.

3. Then find the volume of the prism. V=B×h V = 60 × 10 V = 600

Use the formula for the volume of a prism. Substitute 60 for the area of the base and 10 for the height of the prism. Multiply.

The volume of the triangular prism is 600 cubic centimeters. GO ON Lesson 4-8 Volume of Triangular Prisms and Cylinders

119

YOUR TURN! What is the volume of the triangular prism whose base is a right triangle?

DN DN

1. Use the Pythagorean Theorem to find the length of the other leg of the triangular base of the prism. a2 + b2 = c2 a2 +

16

2

=

20

DN

2

B

a2 + 256 = 400 a2 = 144 a=

12

2. Next, find the area of the base. 1 bh A = __ 2 1 × 12 × 16 A = __ 2 A = 96 3. Then find the volume of the prism. V=B×h V = 96 × V = 864

9

The volume of the triangular prism is 864 cubic centimeters.

What is the volume of the triangular prism whose base is a right triangle?

DN

DN

Nicole

Pedro

7x10 V = 2 × 15 = 375 The volume is 375 cubic centimeters

V = (7 × 10) ÷ 2 × 15 = 525 The volume is 525 cubic centimeters

_

Circle correct answer(s). Cross out incorrect answer(s). 120


DN

Todd V = 7 × 10 × 15 = 1,050 The volume is 1,050 cubic centimeters


Who is Correct?

Guided Practice Name each solid figure. 1

cylinder

triangular prism

2


JO

Find the volume of the solid figure whose base is an isosceles triangle. Step 1 Find the length of the base and height of the triangle. of Use the Pythagorean Theorem to find the height the triangle.

JO


The base of the triangular prism is an isosceles triangle. The length of the base of the isosceles triangle is 6 inches. Therefore, the length of half the base is 6 ÷ 2 = 3 inches. for a and 5 for c in the Substitute for 3 Pythagorean Theorem. a2 + b2 = c2 3 2 + b2 = 5 9 + b2 = 25 b2 = 16 b= 4

2

Remember that the Pythagorean Theorem states that in a right triangle a2 + b2 = c2.

JO

JO

JO

JO JO

Step 2 Find the area of the base of the triangular prism. Use the formula for the area of a triangle. The length of the base of the triangle is and the height of the triangle is

JO

4

6 inches,

inches.

Step 3 Substitute the area of the base and the height into the formula for the volume. Then multiply. The volume of the triangular prism is 144 cubic inches.

1 bh A = __ 2 1× A = __ 2 A=

6 × 4

12

V=B×h V = 12 × V = 144

12

GO ON

Lesson 4-8 Volume of Triangular Prisms and Cylinders

121

Find the volume of each solid figure. Use 3.14 for π. 4

The diameter of the circle is 20 centimeters, so the radius is 20 ÷ 2 = 10 . Use 3.14 for π. A = πr2 A ≈ 3.14 × A ≈ 314

10

DN

DN

2

Substitute the area of the base and the height into the formula for the volume. Then multiply. V=B×h V ≈ 314 × V ≈ 1,570

5

The volume of the cylinder is about 1,570 cubic centimeters. 5

The length of the base of the triangle is of the triangle is 3 yards. 1 bh A = __ 2 __ A=1× 2 A=

6

yards and the height

ZE

6

×

3

9

ZE

ZE

Substitute the area of the base and the height into the volume formula. Then multiply.

13

The volume of the triangular prism is 117 cubic yards. 6

7

DN

N DN

N

N

The volume of the triangular prism is 6 cubic meters.

122


The volume of the cylinder is about 141,300 cubic centimeters.


V=B×h V= 9 × V = 117


Problem-Solving Strategies

Solve. 8

CONTAINERS Marta has a cylinder-shaped container. The diameter of the base is 14 centimeters, and the height is 25 centimeters. What is the volume of the container? Understand

Draw a diagram. Look for a pattern. Act it out. ✓ Use a formula. Work backward.

Read the problem. Write what you know. The diameter of the base of the container is 14 centimeters. The height of the container is 25 centimeters. Pick a strategy. One strategy is to use a formula.

Plan

Think of the problem as two parts. Use a formula to find the area of the circular base. Then use a formula to find the volume. Use 3.14 for π. The diameter of the circle is 14 7 . radius is 14 ÷ 2 =

Solve

centimeters, so the

DN

DN

2


A = πr A ≈ 3.14 × A ≈ 153.86

7

2

Substitute the area of the base and the height into the formula for the volume. Then multiply. V=B×h 25 V ≈ 153.86 × V ≈ 3,846.5 The volume of the container is about 3,846.5 cubic centimeters. Use a calculator to check your answer.

Check

9

CANDLES Conrad’s Candle Shop has two dozen candles shaped like triangular prisms. The largest candle has a height of 7 inches and a triangular base. The length of the triangular base is 6 inches, and its height is 5 inches. What is the volume of the candle? 105 in3 Check off each step.

✔

Understand

✔

Solve

✔

Plan

✔

Check

GO ON Lesson 4-8 Volume of Triangular Prisms and Cylinders

123

10

JO

SCIENCE The cylindrical beakers in Mr. Peters’s biology class have a diameter of 5 inches and a height of 10 inches. What is the volume of each beaker? Use 3.14 for π.

196.25 in3

JO

A box of crackers is in the shape of a rectangular prism. Use what you know about finding the volume of a triangular prism to find the volume of the box.

11

Find the area of the base of the rectangular prism. The area of a rectangle is A = l × w = 4 × 3 = 12

JO

square inches. Multiply the area of the base by the height of the prism: 12 × 6 = 72 cubic inches.

JO JO

JO

JO

Skills, Concepts, and Problem Solving Name each solid figure. 12

13

triangular prism

Find the volume of each solid figure. Use 3.14 for π. 14

15 GU JO

GU GU

JO

The volume of the triangular prism 45 is cubic feet.

124


The volume of the cylinder is about 3,416.32 cubic inches.


cylinder

Solve. 16

CONSTRUCTION To keep the wind from blowing his door shut, Lorenzo made a doorstop. The doorstop is shaped like a triangular prism. The base triangle has a length of 6 centimeters and a height of 15 centimeters. The height of the prism is 8 centimeters. What is the volume of Lorenzo’s doorstop?

360 cm 3 17

FOOD Belinda’s mother stores bagels in a cylinder-shaped container to keep them fresh. The diameter of the container is 6 inches, and its height is 11 inches. What is the volume of the container? Use 3.14 for π.

310.86 in 3


19


20

21

A(n) triangular prism is a prism whose bases are triangular with parallelograms for sides.

Volume

is the number of cubic units needed to fill a three-dimensional figure or solid figure.

cylinder A(n) is a three-dimensional figure having two parallel congruent circular bases and a curved surface connecting the two bases. Writing in Math Explain how to find the volume of a triangular prism.

Find the area of the base using the formula for the area of a triangle. Substitute the area of the base and height into the formula for the volume. Then multiply.

Spiral Review 22

SPORTS The radius of Phoebe’s circular-shaped swimming pool is 5.5 feet. She swam across the pool 4 times. How many feet did Phoebe swim? (Lesson 4-7, p. 109)

44 feet

GO ON

Lesson 4-8 Volume of Triangular Prisms and Cylinders

125

Are the figures congruent? 23

(Lesson 4-5, p. 91)

9

.

¡

¡

7

¡ ¡ -

¡

¡

/

8

congruent LMN and VWX are because corresponding sides and corresponding angles are congruent .

Draw a figure with the description given. 24

right, isosceles triangle

5×3×4

"

#

%

126


$

(Lesson 3-4, p. 27)


Draw a net for the dimensions given.

!

trapezoid

25

26

(Lesson 4-3, p. 73)

Chapter

4

Progress Check 4


Identify the length of the radius and diameter of each circle. 6MG1.2 1

3.5 radius: 7 diameter:

in.

2.75 cm radius: 5.5 diameter: cm

2

in.

DN JO

3

Find the circumference and area of the circle. Use 3.14 for π. 6MG1.2 The circumference of the circle is about 56.52 yards, and the area of the circle is about 254.34 square yards.

4

ZE

_

Find the circumference and area of the circle. Use 22 for π. 6MG1.2 7 The circumference of the circle is about The area of the circle is about

44 or 6_ 2 _ 7

7

44 or 6_ 2 _ 7

7

feet.

square feet.

GU

Find the volume of each solid figure. Use 3.14 for π. 5MG1.3, 6MG1.3


5

6

JO

N

N

JO

N

The volume of the triangular prism 75 is cubic meters.

The volume of the cylinder is about 4,615.8 cubic inches.

Solve. 5MG1.3, 6MG1.3, 6MG1.2 7

FOOD Kizzie purchased a triangular wedge of cheddar cheese. The cheese had a triangle base with length 7 centimeters and height 12 centimeters. The height of the cheese wedge was 9 centimeters. What 3 was the volume of the wedge of cheese? 378 cm

8

BASEBALL According to baseball regulations, the pitcher’s mound, which is circular, must have a diameter of 18 feet. What is 2 the area of a pitcher’s mound? 254.34 ft Lesson 4-8 Volume of Triangular Prisms and Cylinders

127

Chapter

Study Guide

4

Vocabulary and Concept Check cylinder, p. 117

Write the vocabulary word that completes each sentence.

degree, p. 63

1

A(n) protractor is an instrument marked in degrees, used for measuring or drawing angles.

2

A(n) degree angles.

3

A(n) diameter is a chord that passes through the center of a circle.

diameter, p. 109 hypotenuse, p. 99 parallel lines, p. 54 pi (π), p. 109 protractor, p. 63 Pythagorean Theorem, p. 99 4

radius, p. 109 trapezoid, p. 73 triangular prism, p. 117

5

is a unit for measuring

The Pythagorean Theorem states that the sum of the squares of the lengths of the legs in a right triangle is equal to the square of the length of the hypotenuse.

Pi (π)

is the ratio of the circumference of a circle to the diameter of the same circle. Its value is 22 . approximately 3.14 or ___ 7

Label each diagram below. Write the correct vocabulary term in each blank.

parallel lines

7

radius

8

cylinder

9

trapezoid

10

hypotenuse

11

triangular prism

128



6

Lesson Review

4-1

Lines

Example 1

(pp. 54–62)

Name each line. Identify the relationships.

Name each line. Identify the relationships.

:

12

9 "

1

#

2

:

;

perpendicular

The lines are ⊥ lines. PQ 13

&

'

-

.

parallel

The lines are lines. EF

4-2

. YZ

Angles

(pp. 63–71)

;

105° . ∠XYZ measures ∠XYZ is a(n) obtuse angle.

Example 2 Measure and identify the angle. Place the center of the protractor at point B. with the Line up BC line that extends from 0° to 180° on the protractor.

.

15

#

#

35° . ∠MNP measures acute ∠MNP is a(n) angle.

"

1

$

/

"


9

:

intersects XY . AB

LM .

Measure and identify the angle. 14

. The other line is XY . One line is AB The lines cross, or intersect. The lines do not form right angles, so they are not perpendicular lines. The lines are intersecting lines.

$

Look at point C. Read the measure of the passes through the inner angle where BA scale. ∠ABC measures 45°.


129

4-3

Triangles and Quadrilaterals

(pp. 73–80)

Identify each figure.

Example 3

16

Identify the figure.

The figure is a(n)

trapezoid

.

17

The figure has three sides and three angles. The figure is a triangle. None of its sides are equal in length. The figure is a scalene triangle. The figure is a(n)

acute, isosceles triangle

4-4

Add Angles

One angle is an obtuse angle. The figure is an obtuse, scalene triangle.

.

(pp. 81–89)

What types of angles are shown?

Example 4 (

18 %

What is the measure of the missing angle? ¡

¡

&

'

'

3

4 ¡

&

¡

2


19

¡

5

Find the sum of the measures of known angles.

: ¡

¡

¡

m∠Q + m∠R + m∠S = 300° 8

¡

¡

;

The measure of the missing angle is 145° . #

20

¡

"

¡

¡

$

The measure of the missing angle is 30° 130

The sum of the measures of the angles of a quadrilateral is 360°. Subtract the sum of the known measures from 360°. 360° - 300° = 60°


.


See TWE margin.

Sketch complementary angles when one angle’s measure is 80°.

Example 5 Sketch supplementary angles when one angle’s measure is 50°.

21

¡ ¡

¡

¡

1. The sum of supplementary angles equals 180°. Using your protractor, sketch a 180° angle. 2. Find the given angle, 50°, inside the 180° angle and make a dot. Use a straightedge to draw the ray that connects the vertex and the dot at 50°. 3. Find the missing angle by using the protractor’s scale. The supplementary angle is 130°.


4-5 22

Congruent Figures

(pp. 91–98)

Example 6

Are the triangles congruent?

Are the quadrilaterals congruent?

-

/

¡

6 ¡

6

.

7

¡

-

¡

¡

¡

¡

.

¡

¡

5

¡ 4

Corresponding angles are congruent and corresponding sides are congruent. Therefore, the two triangles are congruent.

¡

¡

¡

5

Are corresponding angles congruent? m∠S m∠J. m∠T m∠K. m∠U m∠L. m∠V m∠M.

4

,

¡ +

Are corresponding sides congruent? −− −− ST JK. −− −− TU KL. −−− −−− UV LM. −− −− SV JM.

Quadrilaterals STUV and JKLM are congruent because all of the corresponding angles and sides are congruent. Chapter 4 Study Guide

131

4-6 23

Pythagorean Theorem

Find the length of the leg of the right triangle.

(pp. 99–107)

Example 7

B

Find the length of the leg of the right triangle. From the figure, you know a = 15 and c = 25.

The length of the leg is 24

8

units.

Find the length of the hypotenuse of the right triangle.

4-7 25

Circles

5

units.

What is the area of the circle? 22 for π. Use ___ 7 22 A ≈ ___ × 42 7

What is the area of the circle? JO

7

2

What is the area of the circle? 22 for π. Use ___ 7 22 A ≈ ___ × 62 7 22 × 36 A ≈ ___ 7

132

7

2


GU

The radius is 7 cm; r = 7. 22 Substitute 7 for r and ___ 7 for π in the area formula.

DN

A = πr2 22 × 72 A ≈ ___ 7 22 A ≈ ___ × 49 7 A ≈ 22 × 7 A ≈ 154 The area of the circle is about 154 cm2.


352 in. _ A≈

792 ft _

The length of the leg is 20 units.

Example 8

7

A≈

C

(pp. 109–116)

22 × 16 A ≈ ___

26

Substitute the values of a and c in the Pythagorean Theorem and solve for b. a2 + b2 = c2 152 + b2 = 252 225 + b2 = 625 b2 = 400 b = 20

D


4-8 27

Volume of Triangular Prisms and Cylinders (pp. 117–126)

What is the volume of the triangular prism whose base is a right triangle? JO

JO

JO

The volume of the triangular prism 3 is 420 in . 28

What is the volume of the triangular prism? DN

DN

DN

The volume of the triangular prism is 3 about 600 cm .


29

Example 9 What is the volume of the triangular prism whose base is a right triangle? The length of the base of the triangle is 6 cm, and the height of the triangle is 4 cm.

DN

DN

DN

Use the area-of-atriangle formula to find the area of the base. 1 bh A = __ 2 1×6×4 A = __ 2 A = 12 Substitute the area of the base and the height into the volume formula. Then multiply. V=B×h V = 12 × 10 V = 120 The volume of the triangular prism is 120 cm3.

What is the volume of the cylinder? JO

Example 10 What is the volume of the cylinder?

JO

The volume of the cylinder is about 141.3 in3 . 30

What is the volume of the cylinder? DN

The radius of the circle is 1 ft. Use the formula for the area of a circle to find the area of the base. Use 3.14 for π.

GU

GU

A = πr2 A ≈ 3.14 × 12 A ≈ 3.14 Substitute the area of the base and 3 for the height into the volume formula. Then multiply.

DN

The volume of the cylinder is about 942 cm3 .

V=B×h V ≈ 3.14 × 3 V ≈ 9.42 The volume of the cylinder is about 9.42 ft3. Chapter 4 Study Guide

133

Chapter

Chapter Test

4

Draw each figure. 5MG2.1 1

perpendicular lines OP and XY

2

4

9 0

:

-

.

2

3

5MG2.1

Measure and identify the angle. 5MG2.1 4

"

#

The lines are LM

parallel QR

5MG2.1

rhombus

¡ $

75 ∠ABC measures acute angle ∠ABC is a(n)

lines.

Draw a figure with the description given. 5

5

1


line segment ST

°. .

Draw each type of angle given. 6

5MG 2.1, 5GM2.2

complementary angles

5

4

Find the measure of the missing angle. 7

5MG2.2, 6MG2.2

,

Are the figures congruent?

7MG3.4

8

¡

¡ 4 +

¡

(

5

¡

-

The measure of the missing angle 51 is °.

¡

¡ ¡

6

)

¡

¡

'

STU and FGH are

congruent

because corresponding sides and corresponding congruent angles are . GO ON

134

Chapter 4 Test


3

Find the length of the hypotenuse of the right triangle. 7MG3.3 9


15

D

units.

Use the circle for Exercises 10–13. Use 3.14 for π. 6MG1.2

50 m

10

diameter =

11

radius =

12

circumference ≈

13

area ≈

14

N

Find the volume. Use 3.14 for π.

The volume of the cylinder 3 is about 231.732 ft .

6MG1.3 GU

25 m

GU

157 m

1,962.50 m

Solve. 5MG2.1, 7MG3.4, 6MG1.3 15

CONSTRUCTION Marcel had to fit two pieces of crown molding together at a 90° angle. What type of angle did the molding form?


right

16

COOKING Annie made some fried corn tortillas like the one pictured at the right. What type of triangle do Annie’s tortillas equilateral triangles represent?

17

SHAPES Miss Rustin is making a collage for her club. The two shapes she is using are shown at the right. Are these shapes congruent?

TJODIFT

No, they are not congruent. 18

FOOD The soup kitchen uses cans of soup that have a diameter of 8 inches and a height of 6 inches. What is the volume of a cylindrical soup can? Use 3.14 for π.

about 301.44 in3 Correct the mistakes. 7MG3.3 19

When Aida took her geometry test, this was how she answered the following question: Mr. Hauser has a ladder resting against his bedroom wall. The top of the ladder touches the wall at a height of 12 feet. The ladder is 15 feet in length. How far is the bottom of Mr. Hauser’s ladder from the base of his bedroom wall?

n 5HEDISTANCE OFTHELADDER FROMTHEBASE OFTHEWALLIS FT

See TWE margin. Chapter 4 Test

135

Chapter

4

Standards Practice

Choose the best answer and fill in the corresponding circle on the sheet at right. 1

Which best describes these lines?

5MG2.1

5

Use the Pythagorean Theorem to find the triangle’s missing side length.

7MG3.3

A perpendicular

C bisecting

B intersecting

D parallel

D

ZE

ZE

2

What is the approximate measure of this angle? 5MG2.1

6

F 45°

H 120°

G 90°

J 175°

A 11 yd

C 17 yd

B 15 yd

D 18 yd

What is the circumference of the circle? Use 3.14 for π. 6MG1.2

JO

3

What is the measure of x in the figure?

5MG2.2

H 64 in.

G 50.24 in.

J 200.96 in.

¡

A 30°

C 60°

B 45°

D 150°

7

4

Find the area of the circle. Use 3.14 for π. 6MG1.2

GU

If the two parallelograms are congruent, what is the measure of x? 7MG3.4 Y

A 9.42 ft2 ¡

B 18.84 ft2

¡

C 28.26 ft2

136

F 67°

H 180°

G 113°

J 360°


D 113.04 ft2 GO ON


Y

F 9.14 in.

8

Gavin has a container with the following dimensions. What is its volume? Use 3.14 for π. 6MG1.3

11

What is measure of ∠a and ∠b? 3MG1.2

DN

¡

DN

F 60 cm3 G 188.4 cm3

¡

¡ B

D

¡

E

C

A 135°, 45°

C 60°, 75°

B 135°, 75°

D 45°, 75°

ANSWER SHEET Directions: Fill in the circle of each correct answer.

H 300 cm3 J 942 cm3

9

Find the length of the line to the nearest centimeter. 2MG1.3


A 3 cm B 5 cm C 8 cm D 12 cm

10

1

A

B

C

D

2

F

G

H

J

3

A

B

C

D

4

F

G

H

J

5

A

B

C

D

6

F

G

H

J

7

A

B

C

D

8

F

G

H

J

9

A

B

C

D

10

F

G

H

J

11

A

B

C

D

What is the volume of the solid figure?

3MG1.2

F 12 units3

Success Strategy If you do not know the answer to a question, go on to the next question. Come back to the problem, if you have time. You might find another question later in the test that will help you figure out the skipped problem.

G 24 units3 H 30 units3 J 48 units3 Chapter 4 Standards Practice

137

Index A angle acute, 63–71 complementary, 81–87 corresponding, 89–96 right, 63–71 obtuse, 63–71 straight, 63–71 supplementary, 81–87

D degree angle measure, 63–71

Step-by-Step Problem Solving Measurement and Geometry, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117

diameter, 109–116

F

N net, 27–35

face, 27–35

P

Answer sheet, 51, 137 area parallelogram, 11–17 rectangle, 4–10 triangle, 19–26 Assessment, 48–49, 132–133

G grid paper, 4–14, 16–21, 23–30, 32, 33, 36, 42–46, 48, 66, 67, 69, 76, 78, 80, 83, 85, 88, 90, 93, 96, 98, 116, 126, 129, 134

C California Mathematics Content Standards, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117 Chapter Preview, 3, 53 Chapter Test, 48–49, 134–135 chord, 109–116

circumference, 109–116

coordinate grid, 59 Correct the Mistakes, 49, 135 corresponding angles, 91–98 corresponding sides, 91–98 cube, 37–42 cubic unit, 37–42 cylinder volume, 117–126

H hypotenuse, 99–107

I intersecting lines, 54–62

Index

pi, 109–116 Problem-Solving, see Step-byStep Problem Solving Progress Check, 18, 36, 72, 90, 108, 127 protractor, 63–67, 69, 76, 78, 85, 88, 90, 126, 129, 135 Pythagorean Theorem, 99–105

Q

K Key Concept, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117

quadrilateral area, 73–80 sum of angles, 81–89 types of, 73–80

L R

legs of a triangle, 99–107 line intersecting, 54–62 parallel, 54–62 perpendicular, 54–62

M Manipulatives protractor, 63–67, 69, 76, 78, 85, 88, 90, 126, 129, 135 ruler, 3, 26, 42 Mathematical Reasoning, see

138

perpendicular lines, 54–62 point, 54–62

complementary angles, 81–87 congruent angles, 73–80 figures, 91–98

parallelogram, 19, 73 area 11–17

radius, 109–116 ray, 54–62 Real-World Applications advertising, 106 agriculture, 86 architecture, 80 art, 8, 15, 34, 79, 86 astronomy, 115 badges, 97 baking, 49 baseball, 68, 127 basketball, 7 billiards, 68


circle area, 109–116 circumference, 109–116 diameter, 109–116 radius, 109–116

parallel lines, 54–62


candles, 123 cars, 113 collections, 42 construction, 8, 41, 42, 61, 105, 125, 135 containers, 123 cooking, 135 costumes, 24 cross-country, 88 decorating, 35 design, 18, 33, 59 distance, 107 doors, 9 farming, 16 fences, 71 fitness, 17 flags, 78, 96, 108 food, 10, 18, 25, 116, 125, 127, 135 gardening, 10 geometry, 33 hobbies, 15, 61, 78, 114, 115 home-improvement, 105 horses, 25 houses, 40 jewelry, 34 letters, 89 maps, 58 music, 42 office, 95 packages, 10 packaging, 41, 62 parties, 17 pets, 49 photos, 9 polygons, 97, 107 presents, 32 rectangles, 108 repairs, 90 safety, 77 sailing, 70 school project, 35 science, 124 sculpture, 88 shapes, 135 shopping, 79 signs, 36 sports, 86, 114, 125 storage, 36 tables, 35 tile, 98 time, 69, 72 travel, 107 triangles, 90, 96 volunteering, 23 weather, 26

rectangle, 4, 73–80 area, 4–10 sum of angles, 81–89 rectangular prism surface area, 27–35 volume, 37–42 Reflect, 8, 16, 24, 33, 41, 59, 69, 78, 87, 96, 106, 114, 124 rhombus, 73–80 right angle, 54, 63–71, 99–107 ruler, 3, 26, 42

S Spiral Review, 10, 17, 26, 35, 42, 62, 71, 80, 89, 98, 106, 116, 125 square area, 73–80 sum of angles, 81–89 square number, 99–107 square root, 99–107 square unit, 4–10, 11–17, 19–25 Standards Practice, 50–51, 136–137 Step-by-Step Practice, 6, 13, 22, 30, 39, 57, 66, 76, 83, 94, 103, 112, 121 Step-by-Step Problem Solving Practice, 7–8, 15, 23–24, 32–33, 40–41, 58, 68, 77–78, 86, 95–96, 105, 113–114, 123 Act it out, 95 Draw a diagram, 32, 86, 105 Use a diagram, 58, 68, 77 Use a formula, 7, 15, 23, 113, 123 Use a model, 40

T trapezoid, 73–80 triangle area, 19–26 sum of angles, 81–89 types of, 73–80 triangular prism volume, 117–125

V vertex angle, 63–71 Vocabulary, 4, 11, 19, 27, 37, 54, 63, 73, 81, 91, 99, 109, 117 Vocabulary and Concept Check, 43, 128 Vocabulary Check, 9, 17, 25, 34, 42, 61–62, 70–71, 80, 88, 97, 107, 115, 125 volume cylinder, 117–125 prism, 37–42, 117–125

W Who is Correct?, 5, 12, 21, 29, 38, 56, 65, 76, 83, 93, 102, 111, 120 Writing in Math, 9, 17, 26, 34, 42, 62, 71, 80, 89, 97, 107, 115, 125

Study Guide, 43–47, 128–133 Success Strategy, 51, 137 supplementary angles, 81–89 surface area prism, 27–35

Index

139

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