Geometry Sampler

WrightGroup.com

Teacher’s Edition The U

niversity of Chicago School M

athematics Project

Teacher’s Edition

The University of Chicago School Mathematics Project

UnCh Sc MPr

VOLUME 1 • CHAPTERS 1–7

V1 Sample Lesson Chapter 5 • Lesson 6Sample Lesson Chapter 5 • Lesson 6

TABLE OF CONTENTS

UCSMP Overview

2 About the Third Edition UCSMP Program

3 Program Components

4 The UCSMP Instructional Approach

7 Implementation of UCSMP

8 UCSMP Grades 6-12 Course Summary

10 Geometry Table of Contents

12 Geometry Chapter 5 Planning Resources

16 Geometry Chapter 5 Opener

18 Geometry Lesson 5.6: Auxiliary Figures and Uniqueness

25 Geometry Chapter 5 Projects

27 Geometry Chapter 5 Summary, Self-Test, and Chapter Review

A History of SuccessTwenty-two years of research and development by the University of Chicago School

Mathematics Project (UCSMP) has produced a Pre-K–12 curriculum that consists of

two vertically articulated programs:

(1) the nation’s leading standards-based Pre-K–6

mathematics program, Everyday Mathematics®,

(2) UCSMP Grades 6–12.

The entire UCSMP Pre K–12 curriculum emphasizes problem solving, everyday

applications, the use of technology, and reading in mathematics, while developing

and maintaining basic skills.

Development & Field Tests Over twenty years of research and development

by UCSMP have produced fi eld-tested curricula. Curriculum experts and classroom

teachers wrote the materials over several years. The texts undergo rigorous fi eld

trials at all stages of development to ensure classroom effi cacy.

Reasons for Third EditionThousands of schools using the earlier editions have seen great success in student

achievement and teaching practices. Through fi eld-testing, the Third Edition has

implemented suggestions from mathematics practitioners in an attempt to tailor the

book more to teacher and student needs.

To help middle and high school students succeed under the pressure of increased

testing, the Third Edition takes a broad-based, reality-oriented, and easy-to-

comprehend approach to mathematics. Each of the courses was examined and

revised to take into account:

• higher expectations for middle school students in algebra, geometry and statistics

• increased number of high school students enrolled in higher-level math classes

• active engagement that results in better understanding and performance

• great strides in calculator and computer technology

• up-to-date applications, relevant data, and recent developments in mathematics

Geometry This book retains key features from previous editions of UCSMP

Geometry, while introducing new approaches and methods to mathematics.

• Relevant applications that highlight the importance of mathematics skills

• Blended approach of traditional geometry with transformations and coordinates

• Developed proofs with geometry formulas, postulates, and defi nitions

• Wide variety of rich problems designed to enhance geometry skills and sustain

algebra skills

• New and powerful technology such as graphing calculators, spreadsheets, and

dynamic geometry system

• Reading mathematics to develop the ability to read and understand technical

material

• Required writing for questions to help students clarify their thinking

• Self-Tests and Chapter Reviews at the end of each chapter

• New instructional features, including Guided Examples, Mental Math, and Quiz

Yourself

2 Call 1-800-648-2970

Program ComponentsEach course in UCSMP Grades 6-12 includes

the following components (except where specifi ed):

Student Edition• Hard cover in one volume

• Online Student Edition included

Teacher’s Edition• Hard cover in two volumes

• Electronic version (eTE) on CD-ROM

Assessment Resources• Chapter Quizzes (2) and Tests (1)

• Comprehensive Tests (available Pre-Transition Mathematics

through Advanced Algebra only)

• Answers or Evaluation Guides for all quizzes and tests;

correlation of SPUR Objectives to Chapter Tests

• Assessment Forms

Teaching Resources• Lesson Masters (one-and two-page practice and review

blackline masters; overprinted answers included)

• Resource Masters (generic teaching aids, copies of all Teacher’s Edition

Warm-ups and Additional Examples, and more)

Technology• Electronic Teacher’s Edition CD-ROM (eTE) (includes links to all

ancillary pages and to Answer Masters and Solution to all

questions in the Student Edition)

• Teacher’s Assessment Assistant (TAA) (CD-ROM with a quiz and test generator)

• Dynamic Geometry Sketches

• Dynamic Algebra Sketches

WrightGroup.com



athematics Project

Teacher’s Edition

ity of

athematics

VOLUME

V1

Teacher’s Edition

of

cs

Teache s n

athematics

WrightGroup.com



athematics Project

Teacher’s Edition

The University of Chicago

cs ct

UnCh


V1

Visit us at WrightGroup.com/UCSMP 3

The UCSMP Instructional ApproachThe Teacher’s Edition is an extensive resource to help you address the individual

needs of students. To help you teach and plan your lessons, it is advisable to

fi rst read the Background section in the Teacher’s Edition that accompanies each

lesson. The Background section provides the rationale for the inclusion of topics or

approaches, provides mathematical background, and makes connections between

UCSMP Grades 6–12 materials.

The notes with each lesson in the Teacher’s Edition provide a variety of teaching

ideas, organized around the following four-step instructional plan:

provides questions for students to work on as you begin class.

provides overall notes on how to teach and enhance the lesson,

including procedures for using the Activities. This section also provides

Notes on Examples and Notes on Questions to highlight important aspects of

specifi c examples and questions. Occasionally included are Note-Taking Tips to

help students study. Additional Examples, parallel to the Examples in the Student

Edition, are included with each lesson for added fl exibility.

includes suggested questions

to be completed as homework, pointing

out those that may be appropriate for extra

credit. This section also provides suggested

assignments for the next lesson, including

reading the lesson and doing the Covering the

Ideas section.

includes Ongoing Assessment

suggestions that give students an opportunity

to informally check their understanding of

concepts at the end of each lesson. These options

for differentiation generally employ a quick oral and/

or written activity. In some cases, the Wrap-Up may

simply suggest things teachers should look for

during a closing class discussion. 146 Linear Equations and Inequalities

Variations of ax + b = c

If an equation has the variable on the right side, as in c = ax + b,

the solution can still be obtained by adding the opposite of b, and

multiplying by the reciprocal of a. The Commutative Property of

Addition also implies that ax + b = c is equivalent to b + ax = c.

For example, the following equations can be solved with the same

major steps.

7 = 3 __ 2 x - 53 −53 + 3 __ 2 x = 7

3 __ 2 x - 53 = 7 7 = −53 + 3 __ 2 x

Questions

COVERING THE IDEAS

1. a. Fill in the Blanks When solving 7t - 57 = 97, fi rst add ?

to both sides. Then ? each side by ? . 57; multiply; 1 __ 7

b. Solve and check 7t - 57 = 97.

2. Steps in solving 73y - 432 = 1,101 are shown here.

Given: 73y - 432 = 1,101

Step 1 73y = 1,533

Step 2 y = 21

a. What was done to arrive at Step 1?

b. What was done to arrive at Step 2?

In 3 and 4, the equation is in the form ax + b = c. Find the values of

a, b, and c.

3. 73y - 432 = 1,101 4. 17 - 4y = 88

5. Multiple Choice How do the solutions to 50x - 222 = 60 and

60 = 50x - 222 compare? A

A They are equal.

B They are opposites.

C They are reciprocals.

D None of these are true.

In 6–13, solve and check the equation. See checks at the right.

6. 6x + 42 = 126 x = 14 7. 31 = 11A - 24 A = 5

8. −20y - 2 = 8 y = – 1 __ 2 9. 18 = 16 + 5B B = 2 __ 5

10. 7 + 3 __ 5 d = −5 d = –20 11. 2.4n - 2.4 = 2.4 n = 2

12. 1.06P + 3.25 = 22.86 13. 200 = 4 - 7 __ 2 m m = –56

P = 18.5

Chapter 3

a = 73; b = –432; c = 1,101 a = –4; b = 17; c = 88

432 was added to each side.

Each side was multiplied by 1 ___ 73 .

t = 22; 7(22) - 57 = 154 - 57 = 97

6. 6(14) + 42 = 84 + 42 = 126

7. 11(5) - 24 = 55 - 24 = 31

8. –20 ( – 1 __ 2 ) - 2 =

10 - 2 = 8

9. 16 + 5 ( 2 __ 5 ) = 16 + 2 = 18

10. 7 + 3 __ 5 (–20) = 7 - 12 = –5

11. 2.4(2) - 2.4 = 4.8 - 2.4 = 2.4

12. 1.06 (18.5) + 3.25

= 19.61 + 3.25 =

22.86

13. 4 - 7 __ 2 (–56) = 4 + 196 = 200

146 Chapter 3

Recommended Assignment

• Questions 1–29

• Questions 30 and 31 (extra credit)

• Reading Lesson 3-5

• Covering the Ideas 3-5

Notes on the QuestionsQuestion 3 Look back at Question 2.

Ask students what has been done to

both sides in Steps 1 and 2 in terms

of a, b, and c. (–b was added to both

sides; both sides were multiplied by 1 __ a .)

Question 4 The question may seem

easy, but for many students it is not.

Questions 6–13 You may wish to ask

students for the values of a, b, and

c in each equation. (For example, in

Question 7, a = 11, b = –24, and

c = 31.) Questions 10–13 are

particularly important so that students

do not think that a, b, and c must be

integers.

Note-Taking TipsHave the students put the following steps

in their notebook to solve problems

involving multi-step equations with

variables on one side:

1. Write an equation and simplify it as

much as possible (until there are

three terms at most).

2. Undo the constant term that is

being added or subtracted from the

side that has the variable.

3. Undo the coeffi cient that is being

multiplied by the variable.

4. If the equation is in simplest form

(x = answer), then check by

replacing the variable with the

answer to ensure that the answer

yields a true equation.

Assignment 3 Assignment 3

Accommodating the Learner

If you feel that students are comfortable

solving equations using these methods,

have them explore the table function on a

graphing calculator to solve the equation

2x + 7 = 29. They should enter 2x + 7 in

the Y1 = ” line. Then use the table function

to fi nd the solution. They may need to set

the table to accept their values for x instead

of using automatic values. Students will

discover that 11 is the value for x that gives

a y value of 29.

3-4

SMP08ALG_NA_TE1_C03_L04.indd 146

5/29/07 3:13:39 PM

b, and c. (–b was added to both

sides; both sides were multiplied by 1__sides; both sides were multiplied by __sides; both sides were multiplied by a.)

Question 4 The question may seem

easy, but for many students it is not.

Questions 6–13 You may wish to ask

students for the values of a, b, and

in each equation. (For example, in

Question 7, a = 11, b = –24, and

31.) Questions 10–13 are

particularly important so that students

do not think that a, b, and c must be

integers.

Note-Taking Tips

Inequalities and Multiplication 161

In 19 and 20, solve the equation. (Lesson 3-5) 19. −4(3x - 1.5) + 7 = 39 20. 2(a + 5) - 3(5 + 1 __ 2 a) = −19 21. Grafton went to the store to buy bottles of soda and bags of

chips for a party. He bought bottles of soda for $1.99 each and bags of chips for $2.99 each. He bought twice as many bags of chips as bottles of soda. After paying with two twenty-dollar bills, he received $0.15 in change. (Lessons 3-5, 3-3) a. Defi ne a variable and write an equation describing the

situation. b. How many bottles of soda and bags of chips did Grafton buy? 22. Multiple Choice How do the solutions to 2x - 111 = 35 and

−35 = 2x + 111 compare? (Lessons 3-4, 2-8, 2-4) B A They are equal. B They are opposites. C They are reciprocals. D none of the aboveIn 23 and 24, a fact triangle is given. Write the related facts and determine the value of x. (Lesson 2-7) 23.

5(x � 1)

��

2 � x

31 24.

6

�

3x � 13

42

25. Tomás drove at an average speed of 61 miles per hour for 3 1 __ 4 hours. About how many miles did he travel? (Previous Course) EXPLORATION

26. A rectangle is 12 units by w units. a. Find the values of w that would make the area of the rectangle greater than 84 square units. w > 7 b. Find the values of w that would make the area of the rectangle less than or equal to 216 square units. w ≤ 18 c. Write a sentence to explain what the inequality 60 ≤ 12w < 108 means in relation to the rectangle. d. Solve the inequality in Part c and explain its meaning. 5 ≤ w < 9, which means the width is at least 5 units, but less than 9 units.

Lesson 3-6

w

12

QY ANSWER

{x: x < 32}

19. x = – 13

___ 6 20. a = –28

5 bottles of soda, 10 bags of chips

198.25 mi

Let p be the number of bottles of soda Grafton buys. Then, 1.99p + 2.99(2p) + 0.15 = 40.

23. 5(x + 1) + (2 - x) = 31, (2 - x) + 5(x + 1) = 31, 31 - 5(x + 1) = 2 - x, and 31 - (2 - x) = 5(x + 1); x = 6

24. 6(3x + 13) = 42, 6 + (3x + 13) = 42, 6 = 42

_______ 3x + 13 , and 3x + 13 = 42

___ 6 ; x = –2

26c. We know the area of the rectangle with length 12 units and width w is at least 60 square units but less than 108 square units.

162 Algebra

Cop

yrig

ht ©

Wrig

ht G

roup

/McG

raw

-Hill

page 2 13. −1.8n < 7.2 14. −7(−4h) ≤ 56

15. 3.2m ≥ −17.28 16. 0.2p ≤ 8.1 - 6.4

17. −8r + 5r > 11 + 4 18. t - 1 __ 4 t < 9

19. 6 + 8 > 7w

3-6B

4 2 420

–6 –5 –4

–5.4

–3 6 87 9

–7 –5 –3

0–2 2 4 6

0 4 8 12

–8 –4

h ≤ 2

m ≥ –5.4 p ≤ 8.5

r < –5t < 12

w < 2

n > –4

Algebra 161

Copyright ©

Wright G

roup/McG

raw-H

ill

Lesson Master3-6B

PROPERTIES Objective C 1. What value of k will make the statement true? If x > y then kx < ky.

2. What inequality results if both sides of 7m ≤ 14 are multiplied by 1 __ 7 ? 3. What should both sides of 2 __ 3 n > 5 __ 6 be multiplied by to get n > 5 __ 4 ?

4. What should both sides of −5p ≤ 3 __ 5 be multiplied by to get p ≥ −3

__ 25 ? REPRESENTATIONS Objective C and FIn 5–19, solve the inequality and graph the solution. 5. 13a > −65 6. −2c > 10

7. − 3 __ 4 (2e - 6e) ≥ 9 8. 24b ≤ 3

9. 30 < 2 __ 5 (d + 4d) 10. − 7 __ 4 f ≤ 9 + 5

11. 15 > −2(3g) 12. 8(−6k) < 90 + 6

Questions on SPUR ObjectivesSee pages 178–179 for objectives.

0–5–9 –7 –5 –3

0 3 6

11 15 19 –11 –8 –5

12

- 52 -1

-4 -2 0 2

18

38

- 18

m ≤ 2 3 _ 2

– 1 _ 5

any negative real number

a > –5c < –5

e ≥ 3b ≤ 1 _ 8

d > 15 f ≥ –8

g > – 5 _ 2 k > –2

Lesson 3-6 161

Ongoing AssessmentGraph each inequality on a number line. 1. x ≤ 4

0 2 4 6 8�8 �6 �4 �2 x

2. a ≥ –2

0 2 4 6�8 �6 �4 �2 a

3. –4 ≤ r ≤ 5

0 1 2 3 4 5 6�5 �4 �3 �2 �1 r

4. x < 1

0 1 2�8 �7 �6 �5 �4 �3 �2 �1 x

5. b > 0

0 1 2 3 4 5 6�2 �1

7 b

Wrap-Up4 Wrap-Up43-6


5/29/07 3:48:08 PM

Lesson

144 Linear Equations and Inequalities

In 1637, the French philosopher and mathematician René Descartes started the practice of identifying known quantities by the letters a, b, and c from the beginning of the alphabet and unknown quantities by the letters x, y, and z from the end of the alphabet. Following the practice of Descartes, when we write “solving ax + b = c” we mean that a, b, and c are known numbers and x is unknown. For example, when a = – 3 __ 2 , b = –53, and c = 7, we obtain the equation– 3 __ 2 x + –53 = 7.

In general, any equation of the form ax + b = c, with a not equal to zero, can be solved in two steps. First add the opposite of b to both sides. Then multiply both sides by the reciprocal of a.

Example 1Solve − 3 __ 2 x - 53 = 7.

Solution

− 3 __ 2 x - 53 = 7 Write the equation.

− 3 __ 2 x - 53 + ? = 7 + ? Add ? to each side. 53; 53; 53

− 3 __ 2 x = ? Simplify. 60

– 2 __ 3 ; – 2 __ 3 ? ( − 3 __ 2 x ) = ? (60) Multiply each side by the reciprocal

of − 3 __ 2 .

x = ? Simplify. –40

Be sure to check your solution by substituting it back into the original equation.

Equations That Require Simplifying FirstEquations are often complicated, but they can be simplifi ed into ones that you can solve.

Solving ax + b = c

Chapter 3

3-4

BIG IDEA An equation of the form ax + b = c can be solved in two major steps.

Descartes was a French scientist, mathematician, and philosopher. His statement, “I think; therefore I am,” is very famous.Source: Discourse on Method

GUIDED

a. Which is greater, 1 __ 3 or

0.33? 1 __ 3

b. Which is greater,

1.4 or 33 __ 22 ? 33 ___ 22

c. Which is greater,

− 5 __ 4 or − 4 __ 5 ? − 4 __ 5

Mental Math

144 Chapter 3

Lesson

3-4

GOALSolve an equation of the form ax + b = c regardless of the values of a, b, and c.

SPUR ObjectivesA Solve and check linear equations of the form ax + b = c.

D Use linear equations and inequalities of the form ax + b = c or ax + b < c to solve real-world problems.

Materials/Resources• Lesson Master 3-4A or 3-4B• Resource Master 39• Graphing calculator

HOMEWORKSuggestions for Assignment• Questions 1–29• Questions 30 and 31

(extra credit)• Reading Lesson 3-5• Covering the Ideas 3-5

Local Standards

Present the following problem.

Bill wants to determine the weight of one of 7 identical packages. He has a balance with a 10-kilogram weight and many 500-gram weights. When he puts 5 of the packages on the scale and three 500-gram weights, it balances the 10-kilogram weight. What does each package weigh? 1.7 kg, or 1,700 g

Warm-Up1 Warm-Up1

Background

The power of algebra is in its generality. The same algorithm that solves an equation of the form ax + b = c also solves variations such as b + ax = c, c = ax + b, or similar situations with subtraction or involving fractions or decimals. Do not expect immediate mastery. Although the algorithm does not change, the equations and their solutions look different enough to present diffi culties for many students.

As students proceed through the rest of the chapter, you will need to advise them about the number of steps you want them to write down when they are solving an equation. It is good to keep in mind that the fundamental goal is getting the solution, not writing all the steps. At fi rst, most teachers ask students to show the addition and multiplication to each side of the equation. As students’ skills increase, writing down these steps can be dropped.

SMP08ALG_NA_TE1_C03_L04.indd 144 5/29/07 3:10:25 PM

144 Chapter 3

Solving ax + b = c 145

Example 2When Val works at the zoo on Saturday, she earns $10.80 per hour. She is also paid $8 for meals and $3 for transportation. Last Saturday she received $83.90. How many hours did she work?

Solution Let h = the number of hours Val worked. In h hours she earned 10.80h dollars. So, 10.80h + 8 + 3 = 83.90.

Next, 8 and 3 are added. The resulting equation has the form ax + b = c. Solve for h. 10.80h + 11 = 83.90 Write the equation.10.80h + 11 + –11 = 83.90 + –11 Addition Property of Equality 10.80h = 72.90 Simplify. 1

_____ 10.80

· 10.80h = 1

_____ 10.80

· 72.90 Multiplication Property of Equality h = 6.75 Simplify.Val worked 6.75 hours.

Check If Val worked 6.75 hours at $10.80 per hour, she earned 6.75 · $10.80, or $72.90. Now add $8 for meals and $3 for transportation. The total comes to $83.90.

Example 3The area of the largest rectangle is 94 square centimeters. What is the value of n?

Solution Write an equation to represent the area. (Hint: You can use the sum of areas or the length and width of the big rectangle.) Then solve. Area of left rectangle + Area of right rectangle = 94

? · ? + ? · ? = 94 Write the equation. 12 + ? = 94 Distributive Property

? n + ? = 94 Simplify. 4; 52 ? n + ? + ? = 94 + ? Addition Property of Equality ? ( ? ) = ? ( ? ) Multiplication Property of Equality n = ? Simplify. Be sure to check your solution.

Lesson 3-4

Zookeepers take care of wild animals in zoos and animal parks. They feed the animals, clean their living spaces, work to keep them healthy, and keep them cool in the summer months.Source: Bureau of Labor Statistics

GUIDED

4

3 (n � 10)

4; 52; –52; –52

1 __ 4 ; 4n; 1 __ 4 ; 4210.5

4n + 40

4; 3; 4; (n + 10)

Lesson 3-4 145

Notes on the LessonIt is important for students to realize that there are only two major steps in solving an equation of the form ax + b = c, as shown in Example 1. Other steps merely involve doing arithmetic.Variations of ax + b = c. You may wish to have students identify a, b, and c in the various forms. You must point out that a is the coeffi cient of the unknown (for which you are solving), b is the constant on the same side of the equation as the unknown, and c is the constant on the other side of the equation. For instance, in 5W + 4 = 14, 4 + 5W = 14, or 14 = 4 + 5W, a = 5, b = 4, and c = 14.

Teaching2 Teaching2

Accommodating the LearnerStudents who are likely to have diffi culty with problems like Examples 2 or 3 will need additional time. Plan for at least one additional day for these students. On the fi rst day, have them practice only those problems like Example 1, and then have them write and study the note-taking tips in this section. You could introduce the other examples on the second day after students have had time to study the note-taking tips.

3-4

Additional ExamplesExample 1 Solve – 2 __ 9 x - 1 = 3.

– 2 __ 9 x - 1 + ? = 3 + ?

? x = ?

? ( – 2 __ 9 x ) = ( – 9 __ 2 ) 4 x = ?

Example 2 When Sam works at a local college on Saturdays, he earns $8.20 per hour.

He is also paid $5.00 for meals and $4.00 for transportation. Last Saturday he received $68.45. How many hours did he work? Sam worked 7.25 hr.Example 3 The area of the rectangle is 78 cm2. What is the value of n?

? · ? = 78 6; 2n + 1

? n + 6 = 78 12

? n + 6 + ? = 78 + ?

? n = 72 12

n = ? 6

2n + 1

6

12; −6; −6

1 1– 2 __ 9 4

– 9 __ 2

–18


5/29/07 3:13:30 PM

4 Call 1-800-648-2970

The Teacher’s Edition includes many additional options for

differentiation to promote universal access, including the following:

Gives teachers hints and

instructional strategies on how to

help English language learners and those with weak vocabulary skills

gain access to key mathematical concepts.

Provides suggestions for adjusting an

example, activity, or discussion to make it more accessible to students who may

be struggling with a particular concept.

Provides suggestions for adjusting

an example, activity, or discussion to make it more challenging.

May be a question, problem, activity,

or outside project that extends a concept.

Clearly, all lessons contain more ideas than can be used in one

class period — and there are many additional ways to teach

each lesson. Depending on the background of students, a

challenging activity in one class could be inappropriately easy

in another. Teachers should use their professional judgment to

select and sequence the activities that are appropriate for the

length of a given class period and the needs of their students.

Teachers who have never used group work, manipulatives, or

technology often assume that they are very time-consuming.

Advanced planning and practice will help with the time

management of these very worthwhile activities.


Extension

Horizontal and Vertical Lines 191

900 - 50x - 900 ≥ 300 - 900 Subtract 900 from each side.

–50x ≥ –600 Collect like terms.

–50x

_____ –50

≤ –600

_____ –50

Divide each side by −50.

Change the sense of the

inequality.

x ≤ 12 Simplify.

For the fi rst 12 days, Matt does not have to pay a

service charge.

Deviation from the Mean

For statistical data in a scatterplot, a horizontal line at the mean can

help to show how the data relate to the average value. The hourly

temperatures in Flagstaff, Arizona, on a June day are shown in the

graph below.

Related to each temperature is its deviation from the mean, which

is the difference between the actual temperature and the mean

temperature.

For example, when h = 15 (3 P.M.), the temperature was 82.9°F,

giving a deviation of 82.9 - 68.1 = 14.8°F from the mean. At h = 3

(3 A.M.), the deviation was 52 - 68.1 = −16.1°F.

Use the table and graph of the temperatures at Flagstaff on a June day.

Notice that the values on the vertical axis begin at 45. The interval

0 < y < 45 is compressed on the graph since there are no data in

that interval.

Lesson 4-2

0

45

50

55

60

65

70

75

80

85

90

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Tem

per

atu

re (

˚F)

Hour

y

x

mean

(continued on next page)

Activity

Lesson 4-2 191

Notes on the ActivityRemind students they can produce the

results seen in this Activity using their

calculator.

1. Press STAT.

2. Press ENTER to select EDIT.

3. If there is any old data in column L1,

use DEL → ENTER for each value.

Using arrow keys do the same for L2.

4. Key in data from the table. In L1 key

in 1–24. After each entry press

ENTER. In L2 key in all temperatures

from the table.

5. Press Window – Use these settings:

Xmin – Use the least hour – in this

case 1.

Xmax – Use the greatest hour – in

this case 24.

Xscl – use 1.

Ymin – Use a whole number that is

less than the lowest temperature;

in this case, 40 is good.

Ymax – Use the same logic but pick

a number greater than the highest

temperature. In this case use 85.

Yscl – use 5.

6. Press GRAPH to view the points on

a graph.

7. Graph the mean line. Press y = and

set Y1 = 68.1.

8. Press GRAPH to view all points plus

the mean line.

ENGLISH LEARNERS

Vocabulary Development

While the equations of horizontal lines

( y = k) and vertical lines ( x = h) are

equations of a line, they are special cases

of other forms of equations for lines that

students will see throughout this course.

The general forms are slope-intercept

form y = mx + b, the point-slope form

y - y1 = m(x - x1), and the standard form

Ax + By = C.

4-2


5/30/07 9:44:35 AM

Example 2A family purchased some neon tetras to put in their new fi sh tank. They looked on the Internet to determine at what temperature to set the tank water. One site wrote to keep the water temperature from 72° to 78°F. A second site wrote 68° to 74° and a third site wrote 73° to 81°. To be safe, at what temperature should the family keep the tank?Solution Let t represent the water temperature. Appropriate temperatures for tetras are those satisfying the following.

72° ≤ t ≤ 78° according to Site 168° ≤ t ≤ 74° according to Site 273° ≤ t ≤ 81° according to Site 3

The family wondered if any temperatures could satisfy all three conditions. The graphs show the three intervals separately.

68˚ 70˚ 72˚ 74˚ 76˚ 78˚ 80˚ 82˚t

Site 1

68˚ 70˚ 72˚ 74˚ 76˚ 78˚ 80˚ 82˚t

Site 2

68˚ 70˚ 72˚ 74˚ 76˚ 78˚ 80˚ 82˚t

Site 3

The best temperatures for the tetras are those that lie in all three intervals. This is the intersection in which the three graphs overlap. The tank should have a temperature satisfying 73˚ < t < 74˚.

68˚ 70˚ 72˚ 74˚ 76˚ 78˚ 80˚ 82˚

Site 1 Site 2 Site 3

Describing the Intervals a ≤ x ≤ bMost people would say that the average body temperature is 98.6°F. This fi gure was arrived at in the 19th century. Recent medical research has established that the mean temperature for healthy people is 98.2°F. However, there is some variability among healthy people. According to the new standard, the normal range varies above or below 98.2° by 1.5°. This means that the normal body temperatures t of healthy people range from 98.2 + 1.5 = 99.7 to 98.2 - 1.5 = 96.7. So 96.7 ≤ t ≤ 99.7.

You can combine 98.2 + 1.5 and 98.2 - 1.5 into one expression using ± notation. Then the interval of normal temperatures is written 98.2˚F ± 1.5˚F. The graph on the next page shows this interval and the temperatures of 129 men and women.

Lesson 4-8

Compound Inequalities, And and Or 229

t96.7˚ 98.2˚ 99.7˚

1.5˚ 1.5˚

A neon tetra can live 10 years or more with the proper conditions. Source: animal-world.com

ENGLISH LEARNERS

Vocabulary DevelopmentThe concept of the union and intersection of two or more sets is sometimes very confusing to students. The more you can relate these ideas to real-world situations within the students’ frames of reference, the more likely they will be to keep the two ideas clear. Use a Venn diagram to illustrate your examples.

Accommodating the LearnerIntroduce the students to the idea of a Venn diagram by using the following example. Let A = {3, 4, 5, . . ., 11} and B = {–4, –2, 0, 2, 4, 6, 8}. Using the Venn diagram given, illustrate A ∩ B by shading the intersection. Repeat the process for A ∪ B.

3 5

7 9

10 11

‒4 ‒2

0 2

468

Notes on the LessonExample 2 These data are authentic. You may have students check to see if they can fi nd Web sites that give these different temperature suggestions.

Additional ExampleExample 2 Anita has three recipes for making brownies. The fi rst recipe makes 4 servings, the second makes 8 servings, and the third makes 12 servings. Anita knows that it is not recommended to scale a recipe either up or down beyond four times its original serving size. If s represents the serving size, then the fi rst recipe serving-size scaled interval is 1 ≤ s ≤ 16, the second recipe serving-size scaled interval is 2 ≤ s ≤ 32, and the third recipe serving-size scaled interval is 3 ≤ s ≤ 48. Which interval satisfi es all three recipes? 3 ≤ s ≤ 16

4-8

Lesson 4-8 229


5/30/07 10:43:50 AM

Lesson 4-5

Solving ax + b < cx + d 211

Solution 3 Use a graph. Write an equation describing each tree height. For the beech tree, h = 12 + 0.5t. For the maple tree, h = 4 + t. Graph each equation and fi nd the point of intersection.

0 2 4 6 8 10 12 14 16 18 20 22

2468

10121416182022242628

Hei

gh

t (f

t)

Year

h

t

beech tree

maple tree

The lines intersect at the point (16, 20). So sixteen years after they were planted, the trees were both 20 feet tall. We are looking for the times when the beech tree was taller than the maple tree. So look for values of t where the beech’s line is above that for the maple. These times lie to the left of the intersection point (16, 20). This is where t < 16. The photo was taken less than 16 years after the trees were planted.

The Addition and Multiplication Properties of Inequality can also be used to solve any inequality of the form ax + b < cx + d. In Guided Example 2, two algebraic solutions are given. Solution 2 involves dividing the inequality by a negative number. Recall that multiplying or dividing an inequality by a negative number reverses the inequality sign.

Example 2Solve 7 - 11x ≥ 4x + 12.

Solution 1

7 - 11x ≥ 4x + 12 Write the inequality.7 - 11x + ? ≥ 4x + 12 + ? Add ? to each side. 7 ≥ 15x + 12 Add like terms. 7 - ? ≥ 15x + 12 - ? Subtract ? from each side. –5 ≥ 15x Simplify. ? ≥ ? ? each side by 15. ? ≥ x Simplify. – 1 __ 3

GUIDED


t Height h (ft)

Number of Years

Beech Tree

Maple Tree

0 12 4

4 14 8

8 16 12

12 18 16

16 20 20

20 22 24

24 24 28

11x; 11x; 11x

12; 12; 12

– 5 ___ 15 ; 15x ___ 15 ; divide

Lesson 4-5 211

Notes on the LessonExample 1 A numerical check is not given for this example because each solution checks the other. So you might ask students how they could check that the answer t < 16 is correct. (First check to see that t = 16 makes each side of the original sentence true and then check to see that a value of t that is less than 16 satisfi es the original inequality.)

Guided Example 2 You might ask students which strategy they prefer. This will reinforce the idea that there is more than one way to solve these inequalities. Do they prefer to keep the coeffi cient of the isolated unknown positive, as in Solution 1? Or do they prefer to isolate the unknown on the left, as in Solution 2? For many students, it makes no difference, but some students may have a defi nite preference.

Teaching2 Teaching2


Ask students to fi nd the error in the following proof that 2 < 1.

Let a = –b, a > 0, and b ≠ 0.2a + b > a + b since a + b = 0 2a > a Subtract b from both sides. –2b > –b Substitute –b for a.

–2b ___ –b < – b ___ –b Divide by –b and change the sense.

2 < 1 Simplify.

4-5

Additional ExamplesExample 1 The Bethel Company needs to ship items ordered by customers online. Quick Delivery charges a monthly service charge of $40 a month and $5 per package. On-Time Delivery charges a monthly service charge of $50 per month and $4.50 per package. Initially Quick Delivery will be the more economical carrier. Due to the large volume of packages needing to be shipped each month, at some point during the month, On-Time Delivery will become the more economical choice. When will On-Time Delivery become the more economical choice? when the number of packages being shipped is greater than 20

Example 2 Solve 8 - 6x ≤ –4x + 12.

Solution 1

8 - 6 x + ? ≤ –4 x + 12 + ?

8 ≤ 2 x + 12

8 - ? ≤ 2 x + 12 - ?

–4 ≤ 2 x

? ≤ x


−2

6x; 6x

12; 12

SMP08ALG_NA_TE1_C04_L05.indd 2 5/30/07 10:18:13 AM224 More Linear Equations and Inequalities

Check 1 Use the TRACE feature to read the coordinates of some points on the line. Check that these satisfy the original equation. For example, our TRACE showed the point with x ≈ 10.5, y ≈ −23.7 on the graph.Does 5(10.5) - 2(−23.7) = 100?

99.9 ≈ 100 Yes. It checks.

The point (10.5, −23.7) is very close to the graph of 5x - 2y = 100.Check 2 Compute the coordinates of a point on the line. For example, when x = 0, 5(0) - 2y = 100.

5(0) - 2y = 100

−2y = 100

y = −50

The TRACE on our calculator shows that the point (0, −50) is on the line.

QuestionsCOVERING THE IDEAS

1. There is one temperature at which Celsius and Fahrenheit thermometers give the same reading: − 40°. Verify that C = − 40, F = − 40 satisfi es both F = 1.8C + 32 and C = 5 __ 9 (F - 32). 2. A person with a head circumference of 23.5 inches wears a size 7 1 __ 2 baseball cap. Verify that C = 23.5 and S = 7 1 __ 2 satisfy the formula S = C - 1 _____ 3 . S = 23.5 - 1

_______ 3 = 7 1 __ 2

3. a. Solve p = 2� + 2w for �. � = p __ 2 - w

b. Fill in the Blanks In Part a you are asked to fi nd a formula for ? in terms of ? and ? . �; p; w c. Check your solution to Part a by substituting values for �, w, and p.

In 4 and 5, solve the equation for y. 4. 8x + y = 20 5. 4x - 8y = -40 y = 1 __ 2 x + 5In 6–9, solve the formula for the indicated variable. 6. r = d __ t for d d = rt

7. S = 180n - 360 for n n = S ____ 180 + 2

8. F = m · a for a a = F __ m

9. A = 1 __ 2 (b1 + b2)h for h h = 2A ______ b1 + b2

Chapter 4

1. 1.8(− 40) + 32 = − 72 + 32 = − 40 and 5 __ 9 (− 40 - 32) =

5 __ 9 (−72) = − 40

3c. Answers vary. Sample answer: let � = 1, w = 2, and p = 6. 6 = 2(1) + 2(2), which checkswith � =

p __ 2 - w

= 6 __ 2 - 2 = 3 - 2 = 1

y = 20 - 8x

224 Chapter 4

4-7

Extension

Have students replace the ? in each equation with the number or expression in a, b, and c. Then have them solve for y.a. −4 b. w c. a + b1. 3y = ?

3y = −12 ; y = −4

3y = 3w; y = w

3y = 3(a + b) ; y = a + b

2. 2x - 6 = ?

2x – 6 = −14; x = −4

2x – 6 = 2w – 6; x = w

2x – 6 = 2(a + b) – 6; x = a + b

3. x __ 2 = ?

x __ 2 = −2; x = −4

x __ 2 = w __ 2 ; x = w

x __ 2 = a + b

_____ 2 ; x = a + b

Recommended Assignment• Questions 1–23• Question 24 (extra credit)• Reading Lesson 4-8• Covering the Ideas 4-8

Notes on the QuestionsQuestion 6 Although the distance-rate-time formula is usually given as d = rt, the rate unit (miles per hour) indicates that the rate is derived by dividing distance by time, so the formula given here is quite appropriate.Question 7 S is the sum of the measures of the angles of a convex n-gon.

Question 8 F is force, m is mass, and a is acceleration.Question 9 This is a formula for the area of a trapezoid.


Algebra 187

Copyright ©

Wright G

roup/McG

raw-H

ill

Questions on SPUR ObjectivesSee pages 245–249 for objectives.

Lesson Master4-7A

SKILLS Objective CIn 1–14, solve for the stated variable. 1. x = 2y - 3 for y 2. 5x + 4y = 20 for y

3. C = 2πr for r 4. 3x + 14y = −7y for x

5. E = mc2 for m 6. 9x + 2y = 3x + 2y for x

7. v = x _ t for t 8. PV = nRT for T

9. Z = w __ 5 + w __ 7 for w 10. S = Pl __ 2 + B for l

11. S = Ph + 2B for B 12. 3x + xy = 9 for y

13. g = 3(x + h) for h 14. S = 2πrh + 2πr 2 for h

y = x _ 2 + 3 _ 2 y = 20 – 5x

_____ 4 or y = 5 − 5 _

4 x

r = C __ 2π x = –7y

m = E __ c2 x = 0

t = x _ v T = PV

__ nR

w = 35

__ 12 Z l = 2(S – B)

_____ P

B = S – Ph

____ 2 y = 9 _ x – 3

h = g _ 3 – x h = S – 2πr2

_____ 2πr or

h = s

___ 2πr − r


5/30/07 10:27:35 AM


452 Using Algebra to Describe Patterns of Change

Chapter

Chapter 7

SKILLS Procedures used to get answers

OBJECTIVE A Evaluate functions.

(Lesson 7-6)

In 1–4, suppose f(x) = 10 - 3x. Evaluate the

function.

1. f (2) 4 2. f (−4) 22

3. f (1) + f (0) 17 4. f (3 + 6) −17

5. If g(x) = (11__6 )x, give the value of g(2). 121____

36

6. If h(x) = 2x3, calculate h(4). 128

7. If f (t) = −8t and g(t) = 6 t, give the value of

f(−1) + g(−2). −4

8. If E(m) = 6m and L(m) = m + 5, fi nd a

value for m for which E(m) < L(m).

OBJECTIVE B Calculate function values in

spreadsheets. (Lesson 7-7)

In 9 and 10, use the spreadsheet below.

1000(1.05)^x

1000

1000+50x

1000x0123

12345

A B C

9. Rani wants to put values of the function

with equation y = 1,000(1.05)x in column

B of the spreadsheet. 9a–c. See margin.

a. What formula can she enter in cell B3?

b. What number will appear in cell B3?

c. What should she do to get values of y in

column B when x = 2, 3, 4, 5, . . ., 10?

10. Olivia wants to put values of the function

with equation f(x) = 1,000 + 50x in

column C of the spreadsheet.

a. What formula can she enter in cell C3?

b. What number will appear in cell C3?

c. What should she do to get values of f(x)

in column C when x = 2, 3, 4, 5, . . ., 10?

PROPERTIES The principles behind the

mathematics

OBJECTIVE C Use the language of

functions. (Lessons 7-5, 7-6)

11. Suppose y = f(x).

a. What letter names the independent

variable? x

b. What letter names the function? f

12. A linear function L is graphed below.

y

x

(0, 6)(1.5, 5)

(3, 4)(4.5, 3)

(6, 2)(7.5, 1)

(9, 0)

(10.5, �1)

a. What is L(3)? 4

b. What is the domain of L?

c. What is the range of L?

d. Find a formula for L(x) in terms of x.

L(x) = − 2 __ 3 x + 6

12c. {6, 5, 4, 3, 2, 1, 0, –1}

ChapterReview7

SKILLS

PROPERTIES

USES

REPRESENTATIONS

10a–c. See margin.

Answers vary. Sample answer: 0

12b. {0, 1.5, 3, 4.5, 6, 7.5, 9, 10.5}

Chapter Review

The main objectives for the chapter are

organized in the Chapter Review under

the four types of understanding this book

promotes—Skills, Properties, Uses, and

Representations.

Whereas end-of-chapter material may

be considered optional in some texts,

in UCSMP Algebra we have selected

these objectives and questions with the

expectation that they will be covered.

Students should be able to answer these

questions with about 85% accuracy after

studying the chapter.

You may assign these questions over a

single night to help students prepare for

a test the next day, or you may assign the

questions over a two-day period. If you

work the questions over two days, then

we recommend assigning the evens for

homework the fi rst night so that students

get feedback in class the next day,

and then assigning the odds the night

before the test because the answers are

provided to the odd-numbered questions

in the Selected Answers at the back of

the book.

It is effective to ask students which

questions they still do not understand

and use the day as a total class

discussion of the material that the class

fi nds most diffi cult.

Resources

• Assessment Resources: Chapter 7

Test, Forms A–D; Chapter 7 Test,

Cumulative Form

452 Chapter 7

Chapter Review7

Technology Resources

Teacher’s Assessment Assistant, Ch 7

Electronic Teacher’s Edition, Ch. 7

Additional Answers

9a. Answers vary. Sample answer:

“= B2*1.05”

9b. 1,050

9c. Replicate the formula in B3 down to B12.

10a. Answers vary. Sample answer:

“= C2 + 50”

10b. 1,050

10c. Replicate the formula in C3 down to C12.

SMP08ALG_NA_TE2_C07_EOC.indd 452

11/12/07 11:06:20 AM

Guided Instruction and Active Learning Easy-to-follow, partially

completed Guided Examples model skills and problem solving, and assist students

as they become independent learners. Students are encouraged to stop periodically

to check their understanding with Quiz Yourself (QY) questions. Activities engage

students to discover ideas within the lesson. Games, in Pre-Transition Mathematics

and Transition Mathematics, enable students to practice important mathematics

skills while gaining confi dence in their mathematical abilities.

New and Powerful Technology The use of technology – including graphing

calculators at all grade levels, geometry systems, spreadsheets, the internet, and

other computer applications – is an essential component of the UCSMP Third

Edition program.

Real-life Applications A major UCSMP feature is that real-life applications

are used to introduce and develop concepts in lessons. These real-world

applications have been brought up to date for the third edition and new applications

have been added. Projects and open-ended activities provide an opportunity to do

research or to draw or build models (often collaboratively).

Multi-dimensions to Understanding UCSMP includes the unique

opportunity to develop the mathematical skills and concepts vital in their everyday

life. The SPUR approach provides students with four dimensions of understanding

so they are able to approach and solve problems in different ways.

The SPUR categorization appears at the end of each chapter with the Chapter

Review questions. The Progress Self-Test for each chapter and the Lesson Masters

are also keyed to the SPUR objectives. The categorization is meant to ensure that

teachers are able to provide students with opportunities to gain a broader and

deeper understanding of mathematics

Olivia wants to put values of the function

1,000 + 50x 50x 50

DIMENSIONS OF UNDERSTANDING

S Skills understanding means knowing a way to obtain a solution.

P Properties understanding means knowing properties which you can apply. (Identify or justify the steps in obtaining answer.)

U Uses understanding means knowing situations in which you could apply the solving of this equation. (Set up or interpret a solution.)

R Representations understanding means having a representation of the solving process or a graphical way of interpreting the solution.

The SPUR approach involes four

dimensions of understanding to

enable students to approach and

solve problems in different ways.

6 Call 1-800-648-2970

Mastery and Review for Improved Student Performance Continuous opportunities for review help students master concepts. Each lesson

begins with Mental Math to provide ongoing practice. Review questions at the

end of each lesson allow students to learn over time. These review questions are

designed to engage students in the lesson concepts from different perspectives

and include four types of questions (CARE).

In addition, students are encouraged to assess their own understanding with an End-

of-Chapter Self-Test correlated to objectives. They can then target specifi c areas for

practice and remediation in the SPUR Chapter Review, which is organized by objective.

Implementation of UCSMP

The University of Chicago School Mathematics Project provides an uninterrupted

curriculum from the primary grades through Grade 12. There is a smooth

development and tight alignment of content across the grades. UCSMP is a fl exible

program which allows schools to offer appropriate mathematics to students,

regardless of their grade level. Students can enter UCSMP at any grade but are

advantaged by having had the previous UCSMP courses. The table below shows

how Everyday Mathematics (EM) and the Third Edition texts of UCSMP Grades 6–12

can be used together beginning at Grade 5.

FOUR TYPES OF QUESTIONS

CCovering the Ideas questions demonstrate student knowledge of the overall concepts of the lesson.

AApplying the Mathematics questions go beyond lesson examples, with an emphasis on real-world problem solving.

RReview questions relate either to previous lessons in the course or to content from earlier courses.

E Exploration questions ask students to explore ideas.

GRADE TOP 10-20% OF STUDENTS

NEXT 50% OF STUDENTS

NEXT 20% OF STUDENTS

REMAINDER OF STUDENTS

5 EM 5 or Pre-Transition Mathematics

6 Transition Mathematics EM 6 or Pre-Transition Mathematics

7 Algebra Transition Mathematics Pre-Transition Mathematics

8 Geometry Algebra Transition Mathematics Pre-Transition Mathematics

9 Advanced Algebra Geometry Algebra Transition Mathematics

10 Functions, Statistics, and Trigonometry Advanced Algebra Geometry Algebra

11 Precalculus and Discrete Mathematics Functions, Statistics, and Trigonometry Advanced Algebra Geometry

12 Calculus (Not available through UCSMP)

Precalculus and Discrete Mathematics

Functions, Statistics, and Trigonometry Advanced Algebra


The Seven Courses for Middle School and High School UCSMP has

developed a seven-year middle school through high school curriculum called UCSMP

Grades 6–12. The seven texts around which these courses are built are:

Pre-Transition Mathematics is intended primarily for students

who are ready for a 6th-grade curriculum. It articulates well with

Grade 5 of Everyday Mathematics, Transition Mathematics, and

UCSMP Algebra. Fractions and percents are particularly emphasized.

There is also a major emphasis on dealing with data and geometry.

Algebra is integrated throughout the text as a way of describing

generalizations, as a language for formulas, and as an aid in solving

simple equations.

Transition Mathematics articulates well

with Grade 6 of Everyday Mathematics and

UCSMP Algebra. The curriculum integrates

applied arithmetic, algebra, and geometry, and

connects all of these areas to measurement,

probability, and statistics. Variables are used

to generalize patterns, as abbreviations in

formulas, and as unknowns in problems,

and are represented on the number line

and graphed in the coordinate plane. Basic

arithmetic and algebraic skills are connected

to corresponding geometry topics. Graphing

calculators are assumed for home use.

Algebra has a scope far wider than most

other algebra texts, with mathematical topics

integrated throughout. In addition to the

contexts provided by statistics, geometry,

and probability, expressions, equations,

and functions are described graphically,

symbolically, and in tables. Graphing

calculators are assumed for home use,

while computer algebra system (CAS)

technology is used in the classroom.

8 Call 1-800-648-2970

Geometry integrates coordinates and transformations

throughout, and gives strong attention to measurement

formulas and three-dimensional fi gures. Work with proof

writing follows a carefully sequenced development of the

logical and conceptual precursors to proof. UCSMP

Geometry assumes that students have a graphing

calculator and access to a dynamic geometry system

(DGS) such as The Geometer’s Sketchpad or Cabri.

Advanced Algebra emphasizes facility with algebraic

expressions and forms, especially linear and quadratic forms,

powers and roots, and functions based on these concepts.

Students study logarithmic, trigonometric, polynomial and

other special functions both for their abstract properties

and as tools for modeling real-world situations.

Technology for graphing and CAS technology is

assumed to be available to students.

Functions, Statistics, and Trigonometry (FST) integrates statistics and algebra concepts,

and previews calculus in work with functions and

intuitive notions of limits. Enough trigonometry

is available to constitute a standard precalculus

course in the areas of trigonometry and circular

functions. Technology is assumed available for

student use in graphing, algebraic manipulation,

modeling and analyzing data, and simulating

experiments.

Precalculus and Discrete Mathematics

(PDM) integrates the background students must

have to be successful in calculus with the discrete

mathematics helpful in computer science. It balances

advanced work on functions and trigonometry, an introduction

to limits, and other calculus ideas with work on number systems,

combinatorics, recursion, and graph theory. Technology is assumed

available for student use in graphing, and algebraic manipulation.


Chapter 1 Points and Lines1-1 Points and Lines as Locations 1-2 Ordered Pairs as Points 1-3 Other Types of Geometry 1-4 Undefi ned Terms and First Defi nitions 1-5 Postulates for Points and Lines in Euclidean Geometry 1-6 Betweenness and Distance 1-7 Using a Dynamic Geometry System (DGS) Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 2 The Language and Logic of Geometry2-1 The Need for Defi nitions 2-2 Conditional Statements 2-3 Converses 2-4 Good Defi nitions 2-5 Unions and Intersections of Figures 82-6 Polygons 2-7 Conjectures Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 3 Angles and Lines3-1 Arcs and Angles 3-2 Rotations 3-3 Adjacent and Vertical Angles 3-4 Algebra Properties Used in Geometry 3-5 Justifying Conclusions 3-6 Parallel Lines 3-7 Size Transformations 3-8 Perpendicular Lines 3-9 The Perpendicular Bisector Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 4 Congruence Transformations4-1 Revelecting Points 4-2 Refl ecting Figures 4-3 Miniature Golf and Billiards 4-4 Composing Refl ections over Parallel Lines 4-5 Composing Revelections over Intersecting Lines 4-6 Translations as Vectors 4-7 Isometries 4-8 Transformations and Music Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 5 Proofs Using Congruence5-1 When Are Figures Congruent? 5-2 Corresponding Parts of Congruent Figures 5-3 One-Step Congruence Proofs 5-4 Proofs Using Transitivity 5-5 Proofs Using Refl ections 5-6 Auxiliary Figures and Uniqueness 5-7 Sums of Angle Measures in Polygons Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 6 Polygons and Symmetry6-1 Refl ection Symmetry 6-2 Isosceles Triangles 6-3 Angles Inscribed in Circles 6-4 Types of Quadrilaterals 6-5 Properties of Kites 6-6 Properties of Trapezoids 6-7 Rotation Symmetry 6-8 Regular Polygons 6-9 Frieze Patterns Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 7 Applications of Congruent Triangles7-1 Drawing Triangles 7-2 Triangle Congruence Theorems 7-3 Using Triangle Congruence Theorems 7-4 Overlapping Triangles 7-5 The SSA Condition and HL Congruence 7-6 Tessellations 7-7 Properties of Parallelograms 7-8 Suffi cient Conditions for Parallelograms 7-9 Diagonals of Quadrilaterals 7-10 Proving That Constructions Are Valid Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 8 Lengths and Areas8-1 Perimeter 8-2 Fundamental Properties of Area 8-3 Areas of Irregular Figures 8-4 Areas of Triangles 8-5 Areas of Quadrilaterals 8-6 The Pythagorean Theorem 8-7 Special Right Triangles 8-8 Arc Length and Circumference 8-9 The Area of a Circle Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 9 Three-Dimensional Figures9-1 Points, Lines, and Planes in Space 9-2 Prisms and Cylinders 9-3 Pyramids and Cones 9-4 Drawing in Perspective 9-5 Views of Solids and Surfaces 9-6 Spheres and Sections 9-7 Refl ections in Space 9-8 Making Polyhedra and Other Surfaces 9-9 Surface Areas of Prisms and Cylinders 9-10 Surface Areas of Pyramids and Cones Projects Summary and Vocabulary Self-Test Chapter Review

10 Call 1-800-648-2970

Table of Contents

WrightGroup.com



athematics Project

Teacher’s Edition


UnCh Sc MPr

VOLUME 1 • CHAPTERS 1–7V1

Teacher’s Edition

The niversity y ofhicagochool chool athematicsrojejectct

UnCChScMMPr

VOVOLUME LUME 1 • CHAP1 • CHAPTETERSRS 1–1–77

WrightGroup.com



athematics Project

Teacher’s Edition


UnCh Sc MPr


V1

Chapter 10 Formulas for Volume10-1 Fundamental Properties of Volume 10-2 Multiplication, Area, and Volume 10-3 Volumes of Prisms and Cylinders 10-4 Volumes of Pyramids and Cones 10-5 Organizing and Remembering Formulas 10-6 The Volume of a Sphere 10-7 The Surface Area of a Sphere Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 11 Indirect Proofs and Coordinate Proofs11-1 Ruling Out Possibilities 11-2 The Logic of Making Conclusions 11-3 Indirect Proof 11-4 Proofs with Coordinates 11-5 The Pythagorean Distance Formula 11-6 Equations of Circles 11-7 Means and Midpoints 11-8 Theorems Involving Midpoints 11-9 Three-Dimensional Coordinates Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 12 Similarity12-1 Size Transformations Revisited 12-2 Review of Ratios and Proportions 12-3 Similar Figures 12-4 The Fundamental Theorem of Similarity 12-5 Can There Be Giants? 12-6 The SSS Similarity Theorem 12-7 The AA and SAS Triangle Similarity Theorems Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 13 Similar Triangles and Trigonometry13-1 The Side-Splitting Theorems 13-2 The Angle Bisector Theorem 13-3 Geometric Means in Right Triangles 13-4 The Golden Ratio 13-5 The Tangent of an Angle 13-6 The Sine and Cosine Ratios Projects Summary and Vocabulary Self-Test Chapter Review

Chapter 14 Further Work with Circles14-1 Chord Length and Arc Measure 14-2 Regular Polygons and Schedules 14-3 Angles Formed by Chords or Secants 14-4 Tangents to Circles and Spheres 14-5 Angles Formed by Tangents and a General Theorem 14-6 Three Circles Associated with a Triangle 14-7 Lengths of Chords, Secants, and Tangents 14-8 The Isoperimetric Inequality 14-9 The Isoperimetric Inequality in Three Dimensions Projects Summary and Vocabulary Self-Test Chapter Review


250A Chapter 5

Proofs Using Congruence

Chapter

5

Pacing (in days)

Average Advanced Block

5-1 When Are Figures Congruent?F Make and justify conclusions about congruent fi gures. 1 0.75 0.5

5-2 Corresponding Parts of Congruent FiguresA Identify and determine measures of parts of congruent fi gures.F Make and justify conclusions about congruent fi gures.

1 0.75 0.5

5-3 One-Step Congruence ProofsF Make and justify conclusions about congruent fi gures. 1 0.75 0.5

QUIZ 1 0.5 0.5 0.25

5-4 Proofs Using TransitivityB Construct equilateral triangles and construct the circle through three

noncollinear points.C Find lengths and angle measures using properties of the perpendicular bisector

and alternate interior angles.G Write proofs using the Transitive Properties of Equality or Congruence.J Use the Perpendicular Bisector Theorem and theorems on alternate interior

angles in real situations.

1 0.75 0.5

5-5 Proofs Using Refl ectionsB Construct equilateral triangles and construct the circle through three


and alternate interior angles.H Write proofs using properties of refl ections.J Use the Perpendicular Bisector Theorem and theorems on alternate interior


1 0.75 0.5

5-6 Auxiliary Figures and UniquenessD Use the Triangle-Sum, Quadrilateral-Sum, and Polygon-Sum Theorems to

determine angle measures.I Tell whether auxiliary fi gures are uniquely determined.K Draw fi gures and auxiliary fi gures to aid proofs.

1 0.75 0.5

QUIZ 2 0.5 0.5 0.25

5-7 Sums of Angle Measures in PolygonsD Use the Triangle-Sum, Quadrilateral-Sum, and Polygon-Sum Theorems to

determine angle measures.E Use the Exterior Angle Theorem to answer questions about angles of triangles.K Draw fi gures and auxiliary fi gures to aid proofs.

1 0.75 0.5

1 0.75 0.5

2 1 0.5

1 1 0.5

12 9 5.5

Chapter Overview

Self -Test

Chapter Review

Test

TOTAL


Teacher’s Assessment Assistant, Ch. 5Electronic Teacher’s Edition, Ch. 5

Local Standards

SMP_TEGEO_C05CO_250A-251.indd 250A 4/11/08 10:46:30 AM

12 Call 1-800-648-2970

Chapter 5 Overview

Chapter 5 Overview 250B

Objective s

S kills LessonsSelf-Test

QuestionsChapter Review

Questions

A Identify and determine measures of parts of congruent fi gures. 5-2 1, 2 1–7

B Construct equilateral triangles and construct the circle through three noncollinear points. 5-4, 5-5 4, 12 8–11

C Find lengths and angle measures using properties of the perpendicular bisector and alternate interior angles. 5-4, 5-5 3, 5, 15 12–16

D Use the Triangle-Sum, Quadrilateral-Sum, and Polygon-Sum Theorems to determine angle measures. 5-6, 5-7 7, 9 17–21

E Use the Exterior Angle Theorem to answer questions about angles of triangles. 5-7 10 22–24

Properties

F Make and justify conclusions about congruent fi gures. 5-1, 5-2, 5-3 6 25–29

G Write proofs using the Transitive Properties of Equality or Congruence. 5-4 8 30–33

H Write proofs using properties of refl ections. 5-5 11 34–36

I Tell whether auxiliary fi gures are uniquely determined. 5-6 13 37–41

Uses

J Use the Perpendicular Bisector Theorem and theorems on alternate interior angles in real situations. 5-4, 5-5 14, 17 42–44

Representations

K Draw fi gures and auxiliary fi gures to aid proofs. 5-6, 5-7 16 45–46

Differentiated Options Universal Access



OngoingAssessment Materials

5-1 pp. 255, 256 p. 254 written, p. 256 cardboard, wallpaper or fabric samples, Escher posters

5-2 pp. 260, 261 p. 259 groups, p. 262 cardboard models of congruent fi gures

5-3 p. 265 groups/quiz, p. 268

5-4 pp. 272, 274 p. 271 groups, p. 276

5-5 pp. 279, 280 pairs, p. 281

5-6 pp. 284, 285 p. 284 written/quiz, p. 287

5-7 pp. 290, 291 p. 290 pairs, p. 295

SMP_TEGEO_C05CO_250A-251.indd 250B 4/11/08 10:47:35 AM

250A Chapter 5


Chapter

5

Pacing (in days)

Average Advanced Block

5-1 When Are Figures Congruent?F Make and justify conclusions about congruent fi gures. 1 0.75 0.5

5-2 Corresponding Parts of Congruent FiguresA Identify and determine measures of parts of congruent fi gures.F Make and justify conclusions about congruent fi gures.

1 0.75 0.5

5-3 One-Step Congruence ProofsF Make and justify conclusions about congruent fi gures. 1 0.75 0.5

QUIZ 1 0.5 0.5 0.25

5-4 Proofs Using TransitivityB Construct equilateral triangles and construct the circle through three


and alternate interior angles.G Write proofs using the Transitive Properties of Equality or Congruence.J Use the Perpendicular Bisector Theorem and theorems on alternate interior


1 0.75 0.5

5-5 Proofs Using Refl ectionsB Construct equilateral triangles and construct the circle through three


and alternate interior angles.H Write proofs using properties of refl ections.J Use the Perpendicular Bisector Theorem and theorems on alternate interior


1 0.75 0.5

5-6 Auxiliary Figures and UniquenessD Use the Triangle-Sum, Quadrilateral-Sum, and Polygon-Sum Theorems to

determine angle measures.I Tell whether auxiliary fi gures are uniquely determined.K Draw fi gures and auxiliary fi gures to aid proofs.

1 0.75 0.5

QUIZ 2 0.5 0.5 0.25

5-7 Sums of Angle Measures in PolygonsD Use the Triangle-Sum, Quadrilateral-Sum, and Polygon-Sum Theorems to

determine angle measures.E Use the Exterior Angle Theorem to answer questions about angles of triangles.K Draw fi gures and auxiliary fi gures to aid proofs.

1 0.75 0.5

1 0.75 0.5

2 1 0.5

1 1 0.5

12 9 5.5

Chapter Overview

Self -Test

Chapter Review

Test

TOTAL


Teacher’s Assessment Assistant, Ch. 5Electronic Teacher’s Edition, Ch. 5

Local Standards

SMP_TEGEO_C05CO_250A-251.indd 250A 4/11/08 10:46:30 AM


250C Chapter 5

Chapter 5 Resource Masters

Resource Master 75 Lesson 5-3

Table of Justifications

Some justifi cations that segments are congruent

Some justifi cations that angles are congruent

Defi nition of bisector:If a fi gure is the bisector of a segment, it divides the segment into two congruent

segments. (Lesson 3-9)

Corresponding Angles Postulate:If lines intersected by a transversal are parallel, then corresponding angles are

congruent. (Lesson 3-6)

Defi nition of midpoint:If a point is the midpoint of a segment, it divides the segment into two congruent


Defi nition of angle bisector:If a ray bisects an angle, then it divides the angle into two congruent angles.

(Lesson 3-3)

CPCF Theorem: If fi gures are congruent, then

corresponding segments are congruent. (Lesson 5-2)


corresponding angles are congruent. (Lesson 5-2)

Segment Congruence Theorem: If segments have equal measures, then the

segments are congruent. (Lesson 5-2)

Angle Congruence Theorem: If the measures of angles are equal, then the angles are congruent. (Lesson 5-2)

Defi nition of circle:If a fi gure is a circle, then its radii are


Vertical Angles Theorem:If angles are vertical angles, then they are


Defi nition of congruence: If a segment is the image of another under

an isometry, then the segment and its image are congruent. (Lesson 5-1)

Defi nition of congruence: If an angle is the image of another under an isometry, then the angle and its image

are congruent. (Lesson 5-1)

Resource Master 73

Additional Examples 3. Given �ABC � �DEF Prove m∠ABC = m∠DEF

Conclusions Justifi cations1. �ABC � �DEF 1. _____

2. m∠ABC � m∠DEF 2. _____

3. m∠ABC = m∠DEF 3. If two angles are congruent, then they have the same measure (_____ Congruence Theorem).

4. Given � BD bisects ∠ABC.

Prove m∠ABD = m∠CBDConclusions Justifi cations

1. _____ 1. Given

2. _____ 2. _____

5. x y

1 2 3 4 z

Given m∠2 = m∠4. Justify the conclusion that x � y. Think What is given? m∠2 = m∠4. How are ∠s 2 and 4 related to each

other? They are corresponding angles.Conclusions Justifi cations

1. m∠2 = m∠4 1. Given

You need a reason that ends “Then the lines are parallel.” Recall the second part of the Corresponding Angles Postulate. The justifi cation is _____. Now fi ll in line 2.

2. _____ 2. _____

Lesson 5-3

Resource Masters Chapter 5

None of the generic resource masters apply to this chapter.


Warm-UpWhen learning to read, children often confuse the letters “b,” “d,” “p,” and “q.”

1. In books, which of these letters are usually congruent to the letter “b”?

2. For each of the letters in Warm-Up 1, name the isometry that maps “b” onto the letter if they are on the same line. Be as specifi c as possible.

Questions 12–14

F. 1

1

G.

1

1

H.

1

1

I. 1

1

Resource Master for Lesson 5-1

Resource Master 70

Ci

ht©

Wi

htG

/MG

Hill

Warm-UpIn 1–4, use the fi gure at the right, in which

F

E

A B

O

C

D

quadrilateral CDEO is the image of quadrilateral FABO under a rotation of 180º, to answer the following questions.

1. �BCO is congruent to _____.

2. Name six pairs of segments that are congruent.

3. Name three pairs of angles that are congruent.

4. Hexagon EOCBAF is congruent to _____.

Additional Examples 1. In the fi gure below, MA � TH. If MT = 25 miles, fi nd AH.

M A T H

By the Segment Congruence Theorem, MA = _____. By the Addition Property of Equality, we can add AT to both sides: MA + AT = _____ + AT _________ = AH Thus, if MT = 25 miles, AH = _____.

2. In the fi gure at the right, A

DC

B

E

G

F

1

∠ABC � ∠EFG and BD bisects ∠ABC. If m∠ABC = 16x – 4 and m∠EFG = 4x + 20, fi nd m∠1.

Lesson 5-2


Resource Master 71

Additional Example 3. Given: �BIG � �CAT. a. List the corresponding angles and sides. b. Sketch a possible situation and mark the congruent angles

and sides.

Solution a. You can use the congruence statement

�BIG � �CATto match the corresponding parts: So ∠BIG � _____, ∠IGB � _____, and ∠GBI � _____. And _____ �

___ CA , _____ �

___ AT , and _____ �

___ TC .

b. Two possible triangles are given below with tick marks to indicate the congruent segment.

I

G

B

T A

C

Lesson 5-2


Resource Masters for Lesson 5-3

Ci

ht©

Wi

htG

/MG

Hill

Resource Master 74

6. A

C

BO

In the fi gure above, ___

AO is the refl ection image of ___

CO over � � � OB . Justify the conclusion that

___ AO �

___ CO .

Solution The justifi cation needs to be a conditional that ends “then the angles are congruent.” The second column of the Table of Justifi cations lists many such justifi cations. You need to fi nd one of these whose antecedent has something to do with a refl ection. Find such a reason and either write it out or simply use its name.

Conclusions Justifi cations1. r

� � OB (

___ CO ) =

___ AO 1. _____

2. ___

AO � ___

CO 2. _____

Extenstion

a

b

�

(6x - 26)̊

(2x + 28)̊

2 31

5 64

Question 19

L K

IJ

Lesson 5-3

Ci

ht©

Wi

htG

/MG

Hill


Warm-UpIn 1–4, make a conclusion using Parts a and b of the given information. 1. a. If a person is in Madrid, Spain, then the person is in Europe. b. If a person is in the El Prado Museum, then the person is in

Madrid, Spain.

2. a. If x2 – 4 = 0, then x = 2 or x = –2.

b. If x = 2 or x = –2, then ⎪x⎥ + 3 = 5.

3. a. 3 _ 7 _ 6 _ 7 = 3 __ 6

b. 1 __ 2 = 3 __ 6

4. a. Every square is a rectangle.

b. If a fi gure is a rectangle, then it is an isosceles trapezoid.

Resource Master for Lesson 5-4Resource Masters for Lesson 5-3

Ci

ht©

Wi

htG

/MG

Hill

Resource Master 72

Warm-UpThe segments

___ AB and

___ CD have the same midpoint M as shown here.

A

C

B

D

M

In 1–3, consider this list of justifi cations: a. Vertical angles have the same measure. b. If two angles have the same measure, then they are congruent. c. Angles in a linear pair are supplementary.

Which statement from the list justifi es the given conclusion?

1. m∠AMC + m∠CMB = 180º

2. m∠AMC = m∠BMD

3. ∠AMC � ∠BMD

Additional Examples 1. Sk is a size transformation of magnitude k. What can you conclude

about ∠MNO and Sk∠MNO?

2. Given: Points A and B are on the circle with center O. What can you conclude about OA and OB?

Lesson 5-3

SMP_TEGEO_C05CO_250A-251.indd 250C 4/11/08 10:47:56 AM

14 Call 1-800-648-2970

Chapter 5 Overview

Chapter 5 Overview 250D

Resource Master 83

Uniquely Determined FiguresThrough Lesson 5-5, there are 11 uniquely defi ned fi gures:

1. Given two points, the segment joining them (Point-Line-Plane Postulate part a)

2. Given a ray and a distance x, the point on the ray at that distance from the endpoint (Point-Line-Plane Postulate part a and Distance Postulate part a)

3. Given a segment, its midpoint (defi nition of midpoint) 4. Given a ray and a measure x between 0º and 180º, an angle with the

ray as one side on a given half-plane of the ray and with measure x (Angle Measure Postulate)

5. Given an angle, its bisector (defi nition of angle bisector and Angle Measure Postulate)

6. Given a line and a point, the perpendicular to that line through that point (uniqueness of angle, defi nition of perpendicular, and Linear Pair Theorem)

7. Given a segment, its perpendicular bisector (defi nition of perpendicular bisector, Distance Postulate, and Angle Measure Postulate)

8. Given a point and a line, the refl ection image of that point (Refl ection Postulate part a)

9. Given a point P and a transformation T, the image point T(P) (defi nition of transformation)

10. Given a point and a line, the perpendicular to that line from that point (Refl ection Postulate)

11. Given three noncollinear points, the circle through them (proved in Example 1, Lesson 5-5)

Lesson 5-6


Additional Examples 1. In the fi gure at the right, a � b.

c

12

3

a

b

If m∠3 = 124, fi nd m∠2 and m∠1.

2. Prove the Alternate Interior Angles

m

n

�

4 31 2

8 75 6

Theorem using the fi gure at the right as a drawing.

Given ∠4 � ∠6 Prove m � n

Conclusions Justifi cations1. ∠4 � ∠6 1. Given

2. ∠6 � ∠8 2. _____

3. ∠4 � ∠8 3. _____

4. m � n 4. If two corresponding angles have the same measure, then the lines are parallel. (Corresponding Angles Postulate)

Application of Activity Algorithm

BA

C

D

Ci

ht©

Wi

htG

/MG

Hill


Summary of Postulates and Theorems about Parallel Lines and Transversals

If given parallel lines, then Ways to prove parallel lines

• Corresponding angles are congruent. (Corresponding Angles Postulate)

• If two lines are cut by a transversal and form congruent corresponding angles (Corresponding Angles Postulate)

• Alternate interior angles are congruent. (Parallel Lines Theorem)

• If two lines are cut by a transversal and form congruent alternate interior angles (Alternate Interior Angles Theorem)

• Alternate exterior angles are congruent. (Parallel Lines Theorem)

• If two lines are cut by a transversal and form congruent alternate exterior angles (Alternate Exterior Angles Theorem)

• Same-side interior angles are supplementary. (Parallel Lines Theorem)

• If two lines are cut by a transversal and form supplementary same-side interior angles (Same-Side Interior Angles Theorem)

Question 10 Question 11

1

34

6

8 7

5

2

Singles (27 ft)

Doubles (36 ft)

21 ft Servicearea

Center mark

Singles sidelineDoubles sideline

3 ft at center

Baseline

78 ft

Alley lineAlle

y lin

e

Source: ©Image Source/SuperStock, p. 79

Resource Master 79

Question 12 Accommodating the Learner

2

43

1

52"

43"

24"

10"

16"

5 812

"

87°

45° 48°Q D R

AU

Question 16Fill in the Blanks Copy and supply the missing t

m

n

23

1

parts in the proof of the following theorem:

Theorem If same-side interior angles are supplementary, then the lines are parallel.

Given ∠1 and ∠2 are supplementary, m∠2 = x.

Prove m � n

Proof It is given that and . Because ∠1 and ∠2 are supplementary, then the measures of ∠1 and ∠2 add to and so, m∠1 + x = . ∠2 and ∠3 are supplementary angles because they are and so, x + m∠3 = . By substitution, m∠1 + x = x + m∠3. So by the Addition Property of Equality and ∠1 and ∠3 are angles. Therefore, m � n by the Postulate, which states that .

Lesson 5-4

Ci

ht©

Wi

htG

/MG

Hill

Resource Master 80

Warm-UpName the two defi ning properties of the perpendicular bisector of a segment.

Additional Examples

A

Y

X�

1. Given rℓ(X) = Y

Prove AX = AYConclusions Justifi cations

1. rℓ(X) = Y 1. Given

2. ℓ is the perpendicular bisector of

__ XY

2. _____

3. AX = AY 3. _____

2. Prove If rm(X) = X, rm(Y) = W, and Y

W

Z mX

rm(Z) = Z, then �XYZ � �XWZ.

Given rm(X) = X, r

m(Y) = W, and r

m(Z) = Z

Prove �XYZ � �XWZProof


m(X) = X, r

m(Y) = W,

and rm(Z) = Z

1. Given

2. rm(�XYZ) = �XWZ 2. _____

3. �XYZ � �XWZ 3. _____

Lesson 5-5 Resource Master 81 Lesson 5-5

Question 5 Question 6

D

B

C

A

Points closest to D

Points closest to B

Points closest to A

Pointsclosestto C

R

S

T

Q

Question 7

N

A

E

M

Ci

ht©

Wi

htG

/MG

Hill


Warm-UpSuppose you wanted to prove the following statement: If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180º, then the lines will intersect on that same side of the transversal. Draw a fi gure and state the Given and To Prove for your fi gure.

Additional Examples 1. Tell if each thing is uniquely determined. If not, why not? a. Given ∠DEF, bisector of ∠DEF b. Given point A on line m, point x on m a given distance from A c. Given line n and point C on n, the perpendicular to n through C

2. In �ABC, a student wished to draw as an auxiliary line the perpendicular bisector of

___ AC that passes through the vertex of ∠B.

Is this always possible? Why or why not?

C

DA

E

B

Resource Masters for Lesson 5-6Resource Master for Lesson 5-5Resource Master for Lesson 5-5

Resource Master for Lesson 5-4Resource Master for Lesson 5-4Resource Master for Lesson 5-4

Ci

ht©

Wi

htG

/MG

Hill

Resource Master 86

Question 3

A

B

C

D

E

FG

123˚

139˚

136˚

101˚114˚

Question 4

A

B

C D

E

1 2 3

4

5 6 7 8

9

Lesson 5-7


Resource Master 85

Additional Examples 1. Use the Triangle-Sum Theorem to discover and prove the

Pentagon-Sum Theorem. Solution Let S = the sum of the measures 1 2 3

76

54

P

T

A

E

N

the angles of the convex PENTA at the right.

S = m∠APE + m∠E + m∠ENT + m∠NTA + m∠ADraw auxiliary segments _____ and _____.

Notice that ___

PT and ___

PN have split angle P into three adjacent angles and angles T and N into two adjacent angles each. Thus, using the Angle Addition Postulate: m∠APE = m∠_____ + m∠_____ + m∠_____, m∠ENT = m∠_____ + m∠_____, and m∠NTA = m∠_____ + m∠_____.

Thus, by substituting these expressions for m∠APE, m∠ENT, and m∠NTA, S = (m∠_____ + m∠_____ + m∠_____) + m∠E + (m∠_____ + m∠_____) + m∠_____ + m∠_____ + m∠A.

Rearrange and regroup the terms in this equation to getS = (m∠1 + m∠A + m∠7) + (m∠2 + m∠5 + m∠6) + (m∠3 + m∠E + m∠4).We know from the Triangle-Sum Theorem thatm∠1 + m∠A + m∠7 = _____, m∠2 + m∠5 + m∠6 = _____, and m∠3 + m∠E + m∠4 = _____.

Consequently, by substitution, S = _____ + _____ + _____ so S = _____.

2. Describe how to calculate the measure of one interior angle in a dodecagon (12-sided polygon) whose angles are all congruent.

Lesson 5-7


Ci

ht©

Wi

htG

/MG

Hill


Warm-UpThis Warm-Up leads directly into Activity 2. 1. Draw a convex polygon with seven sides. Pick one vertex and draw

all the possible diagonals from that vertex. How many triangles are formed?

2. Repeat Warm-Up 1 but start with a different vertex. Does choosing another vertex affect the number of triangles that can be formed?

3. Repeat Warm-Up 1 but with a polygon of eight sides. How many triangles are formed?

4. Generalize Warm-Ups 1–3. How many triangles are formed when all the diagonals from one vertex of a convex n-gon are drawn?

Activity 2A

5-gon

A

6-gon

A

7-gon

A

8-gon

Number of sides 5 6 7 8 … n

Number of triangles formed …

Sum of the measures of the angles in the polygon

…


SMP_TEGEO_C05CO_250A-251.indd 250D 4/15/08 11:38:24 AM

250C Chapter 5

Chapter 5 Resource Masters


Table of Justifications

Some justifi cations that segments are congruent

Some justifi cations that angles are congruent

Defi nition of bisector:If a fi gure is the bisector of a segment, it divides the segment into two congruent


Corresponding Angles Postulate:If lines intersected by a transversal are parallel, then corresponding angles are


Defi nition of midpoint:If a point is the midpoint of a segment, it divides the segment into two congruent


Defi nition of angle bisector:If a ray bisects an angle, then it divides the angle into two congruent angles.

(Lesson 3-3)


corresponding segments are congruent. (Lesson 5-2)


corresponding angles are congruent. (Lesson 5-2)

Segment Congruence Theorem: If segments have equal measures, then the

segments are congruent. (Lesson 5-2)

Angle Congruence Theorem: If the measures of angles are equal, then the angles are congruent. (Lesson 5-2)

Defi nition of circle:If a fi gure is a circle, then its radii are


Vertical Angles Theorem:If angles are vertical angles, then they are


Defi nition of congruence: If a segment is the image of another under

an isometry, then the segment and its image are congruent. (Lesson 5-1)

Defi nition of congruence: If an angle is the image of another under an isometry, then the angle and its image

are congruent. (Lesson 5-1)

Resource Master 73

Additional Examples 3. Given �ABC � �DEF Prove m∠ABC = m∠DEF

Conclusions Justifi cations1. �ABC � �DEF 1. _____

2. m∠ABC � m∠DEF 2. _____

3. m∠ABC = m∠DEF 3. If two angles are congruent, then they have the same measure (_____ Congruence Theorem).

4. Given � BD bisects ∠ABC.

Prove m∠ABD = m∠CBDConclusions Justifi cations

1. _____ 1. Given

2. _____ 2. _____

5. x y

1 2 3 4 z

Given m∠2 = m∠4. Justify the conclusion that x � y. Think What is given? m∠2 = m∠4. How are ∠s 2 and 4 related to each

other? They are corresponding angles.Conclusions Justifi cations

1. m∠2 = m∠4 1. Given

You need a reason that ends “Then the lines are parallel.” Recall the second part of the Corresponding Angles Postulate. The justifi cation is _____. Now fi ll in line 2.

2. _____ 2. _____

Lesson 5-3

Resource Masters Chapter 5

None of the generic resource masters apply to this chapter.


Warm-UpWhen learning to read, children often confuse the letters “b,” “d,” “p,” and “q.”

1. In books, which of these letters are usually congruent to the letter “b”?

2. For each of the letters in Warm-Up 1, name the isometry that maps “b” onto the letter if they are on the same line. Be as specifi c as possible.

Questions 12–14

F. 1

1

G.

1

1

H.

1

1

I. 1

1


Resource Master 70

Ci

ht©

Wi

htG

/MG

Hill

Warm-UpIn 1–4, use the fi gure at the right, in which

F

E

A B

O

C

D

quadrilateral CDEO is the image of quadrilateral FABO under a rotation of 180º, to answer the following questions.

1. �BCO is congruent to _____.

2. Name six pairs of segments that are congruent.

3. Name three pairs of angles that are congruent.

4. Hexagon EOCBAF is congruent to _____.

Additional Examples 1. In the fi gure below, MA � TH. If MT = 25 miles, fi nd AH.

M A T H

By the Segment Congruence Theorem, MA = _____. By the Addition Property of Equality, we can add AT to both sides: MA + AT = _____ + AT _________ = AH Thus, if MT = 25 miles, AH = _____.

2. In the fi gure at the right, A

DC

B

E

G

F

1

∠ABC � ∠EFG and BD bisects ∠ABC. If m∠ABC = 16x – 4 and m∠EFG = 4x + 20, fi nd m∠1.

Lesson 5-2


Resource Master 71

Additional Example 3. Given: �BIG � �CAT. a. List the corresponding angles and sides. b. Sketch a possible situation and mark the congruent angles

and sides.

Solution a. You can use the congruence statement

�BIG � �CATto match the corresponding parts: So ∠BIG � _____, ∠IGB � _____, and ∠GBI � _____. And _____ �

___ CA , _____ �

___ AT , and _____ �

___ TC .

b. Two possible triangles are given below with tick marks to indicate the congruent segment.

I

G

B

T A

C

Lesson 5-2


Resource Masters for Lesson 5-3

Ci

ht©

Wi

htG

/MG

Hill

Resource Master 74

6. A

C

BO

In the fi gure above, ___

AO is the refl ection image of ___

CO over � � � OB . Justify the conclusion that

___ AO �

___ CO .

Solution The justifi cation needs to be a conditional that ends “then the angles are congruent.” The second column of the Table of Justifi cations lists many such justifi cations. You need to fi nd one of these whose antecedent has something to do with a refl ection. Find such a reason and either write it out or simply use its name.


� � OB (

___ CO ) =

___ AO 1. _____

2. ___

AO � ___

CO 2. _____

Extenstion

a

b

�

(6x - 26)̊

(2x + 28)̊

2 31

5 64

Question 19

L K

IJ

Lesson 5-3

Ci

ht©

Wi

htG

/MG

Hill


Warm-UpIn 1–4, make a conclusion using Parts a and b of the given information. 1. a. If a person is in Madrid, Spain, then the person is in Europe. b. If a person is in the El Prado Museum, then the person is in

Madrid, Spain.

2. a. If x2 – 4 = 0, then x = 2 or x = –2.

b. If x = 2 or x = –2, then ⎪x⎥ + 3 = 5.

3. a. 3 _ 7 _ 6 _ 7 = 3 __ 6

b. 1 __ 2 = 3 __ 6

4. a. Every square is a rectangle.

b. If a fi gure is a rectangle, then it is an isosceles trapezoid.

Resource Master for Lesson 5-4Resource Masters for Lesson 5-3

Ci

ht©

Wi

htG

/MG

Hill

Resource Master 72

Warm-UpThe segments

___ AB and

___ CD have the same midpoint M as shown here.

A

C

B

D

M

In 1–3, consider this list of justifi cations: a. Vertical angles have the same measure. b. If two angles have the same measure, then they are congruent. c. Angles in a linear pair are supplementary.

Which statement from the list justifi es the given conclusion?

1. m∠AMC + m∠CMB = 180º

2. m∠AMC = m∠BMD

3. ∠AMC � ∠BMD

Additional Examples 1. Sk is a size transformation of magnitude k. What can you conclude

about ∠MNO and Sk∠MNO?

2. Given: Points A and B are on the circle with center O. What can you conclude about OA and OB?

Lesson 5-3

SMP_TEGEO_C05CO_250A-251.indd 250C 4/11/08 10:47:56 AM


250

Chapter


5

Contents

5-1 When Are Figures Congruent?

5-2 Corresponding Parts of Congruent Figures

5-3 One-Step Congruence Proofs

5-4 Proofs Using Transitivity

5-5 Proofs Using Refl ections

5-6 Auxiliary Figures and Uniqueness

5-7 Sums of Angle Measures in Polygons

Recall that two objects are congruent if they are exactly the same size and shape. Congruent objects are everywhere.

Teachers duplicate worksheets for students,

and businesses photocopy photographs and

diagrams. Tool-and-die makers create molds

(the dies) for cutting and forging metal so

that manufacturers can make identical parts.

Which of the fi gures below do you think are

congruent to Figure A?

BA C

ED F

250 Chapter 5

Chapter

5

PacingEach lesson in this chapter is designed to be covered in 1 day. At the end of the chapter, you should plan to spend 1 day to review the Self-Test, 1 to 2 days for the Chapter Review, and 1 day for a test. You may wish to spend a day on projects, and possibly a day is needed for quizzes. You may want to linger on one or two lessons a little longer. This chapter should therefore take 10 to 12 days. We strongly advise you not to spend more than 13 days on this chapter. Despite the urge you might have to take more time to develop proof competence, the idea of proofs continues to be developed in Chapters 6 and 7.

OverviewThe overview emphasizes some of the places in which congruent fi gures are desired:

• copied documents• manufactured products• designs• packing situations

Because the idea of congruence has been used for thousands of years to deduce the properties of fi gures, we tend to think that congruence is obvious. Yet, as evidenced by the diffi culty students initially have with proofs, the step from seeing congruence in fi gures to using congruence to deduce the properties of fi gures is not obvious. Point out that the major idea in this chapter is to determine how much information we can glean from bits of information given in a particular situation. In this chapter, these bits of information will often involve knowing that some fi gures are congruent.

Using Pages 250–251Refer students to Figures a–f on page 250. Ask, “Have you seen puzzles in which you need to determine whether the fi gures are congruent?” Remind students that we use drawings to determine whether fi gures

Chapter 5 Overview

Chapters 5 –7 combine two approaches to the study of polygons. These approaches use ideas from Chapter 2 (good defi nitions), Chapter 3 (angles formed by parallel lines), and Chapter 4 (refl ections and congruence). In Chapters 5 and 7, students will write proofs. In Chapter 6, those ideas are applied to deduce and help students learn the properties of polygons that possess refl ection symmetry. A global property of a fi gure—its symmetry—is used to deduce some of its specifi c

properties. In Chapter 7, the more traditional approach of triangle congruence is employed to deduce properties of fi gures that do not possess symmetry. In that approach, congruent sides and angles are used to deduce other specifi c properties. Combined, the two approaches give students the ability to understand and deduce the properties of polygons.

This chapter introduces the fundamental ideas of proof that are needed for both approaches and shows their power by

SMP_TEGEO_C05CO_250A-251.indd 250 4/11/08 10:48:23 AM

16 Call 1-800-648-2970

251

Tiled fl oors and bricked driveways can form

interesting patterns. The congruent fi gures

below fi t together nicely to form an attractive

pattern. This is possible because they have

corresponding parts that are congruent.

People who manufacture dice for games

have to make sure the faces of the die are

congruent. Congruence helps assure that

each face has an identical chance of facing

up when the die is randomly tossed.

In this chapter, you will take a close look at

some of the properties of congruent fi gures.

You will see how congruent fi gures are used

and use them yourself to deduce properties

of angles and segments within particular

fi gures. These properties and results have

been key ideas for as long as people have

studied geometry, so in this chapter you will

be introduced to some of the history of the

development of geometry.

Chapter 5 Opener 251

deducing some very important theorems. Lessons 5-1 and 5-2 introduce the basic properties of congruence, including its defi nition, the theorem that corresponding parts of congruent fi gures are congruent, and theorems that reveal relationships between congruence and equality. Lesson 5-3 introduces one-step congruence proofs and general ideas about proofs. Lesson 5-4 shows the power of proofs with the Transitive Property of Equality and deduces the theorems about parallel lines and alternate

interior angles. In Lesson 5-5, refl ection properties help prove the Perpendicular Bisector Theorem and other geometric conclusions. Lesson 5-6 introduces auxiliary fi gures and uniqueness. Lesson 5-7 develops the sum of the measures of the angles of any triangle, which then deduces a formula for the sum of the measures of the interior angles of any convex polygon.

Throughout this chapter, students are asked to draw with a straightedge, a compass, a protractor, and a DGS.

“seem” congruent. If you have not already discussed a defi nition of congruence, point out that the defi nition of congruence in Lesson 5-1 allows fi gures to have different orientations and still be congruent, as in Figures a and b. Lesson 5-1 also introduces the phrase oppositely congruent. Figure c is a rotation image of Figure b, so they are directly congruent. Figure e is a rotation image of Figure a, so they are directly congruent. Figures d and f are not congruent to the others. Notice that the partial tessellation with Figure a as a fundamental region involves congruent fi gures of different orientations.

The 3-dimensional fi gures pictured on page 251 are the fi ve regular polyhedra. These polyhedra have congruent regular polygons as their faces. The congruent faces and their symmetry planes enable us to use any of these polyhedra as dice, though obviously the cube is much more commonly used than are the others.

Chapter 5 Projects

At the end of each chapter, you will fi nd projects related to the chapter. At this time you might want to have students look over the projects on pages 296 and 297. You might want to have students tentatively select a project on which to work. Then, as students read and progress through the chapter, they can fi nalize their project choices.

Sometimes students might work alone. At other times, you might let them collaborate with classmates for a presentation and discussion. We recommend that you allow for diversity and encourage students to use their imaginations when presenting their projects. As students work on projects throughout the year, they should see many uses of mathematics in the real world.


250

Chapter


5

Contents

5-1 When Are Figures Congruent?

5-2 Corresponding Parts of Congruent Figures

5-3 One-Step Congruence Proofs

5-4 Proofs Using Transitivity

5-5 Proofs Using Refl ections

5-6 Auxiliary Figures and Uniqueness

5-7 Sums of Angle Measures in Polygons

Recall that two objects are congruent if they are exactly the same size and shape. Congruent objects are everywhere.

Teachers duplicate worksheets for students,

and businesses photocopy photographs and

diagrams. Tool-and-die makers create molds

(the dies) for cutting and forging metal so

that manufacturers can make identical parts.

Which of the fi gures below do you think are

congruent to Figure A?

BA C

ED F

250 Chapter 5

Chapter

5

PacingEach lesson in this chapter is designed to be covered in 1 day. At the end of the chapter, you should plan to spend 1 day to review the Self-Test, 1 to 2 days for the Chapter Review, and 1 day for a test. You may wish to spend a day on projects, and possibly a day is needed for quizzes. You may want to linger on one or two lessons a little longer. This chapter should therefore take 10 to 12 days. We strongly advise you not to spend more than 13 days on this chapter. Despite the urge you might have to take more time to develop proof competence, the idea of proofs continues to be developed in Chapters 6 and 7.

OverviewThe overview emphasizes some of the places in which congruent fi gures are desired:

• copied documents• manufactured products• designs• packing situations

Because the idea of congruence has been used for thousands of years to deduce the properties of fi gures, we tend to think that congruence is obvious. Yet, as evidenced by the diffi culty students initially have with proofs, the step from seeing congruence in fi gures to using congruence to deduce the properties of fi gures is not obvious. Point out that the major idea in this chapter is to determine how much information we can glean from bits of information given in a particular situation. In this chapter, these bits of information will often involve knowing that some fi gures are congruent.

Using Pages 250–251Refer students to Figures a–f on page 250. Ask, “Have you seen puzzles in which you need to determine whether the fi gures are congruent?” Remind students that we use drawings to determine whether fi gures

Chapter 5 Overview

Chapters 5 –7 combine two approaches to the study of polygons. These approaches use ideas from Chapter 2 (good defi nitions), Chapter 3 (angles formed by parallel lines), and Chapter 4 (refl ections and congruence). In Chapters 5 and 7, students will write proofs. In Chapter 6, those ideas are applied to deduce and help students learn the properties of polygons that possess refl ection symmetry. A global property of a fi gure—its symmetry—is used to deduce some of its specifi c

properties. In Chapter 7, the more traditional approach of triangle congruence is employed to deduce properties of fi gures that do not possess symmetry. In that approach, congruent sides and angles are used to deduce other specifi c properties. Combined, the two approaches give students the ability to understand and deduce the properties of polygons.

This chapter introduces the fundamental ideas of proof that are needed for both approaches and shows their power by



282 Proofs Using Congruence

LessonAuxiliary Figures and Uniqueness

Chapter 5

5-6

BIG IDEA Introducing a line or part of a line into a fi gure can be a helpful strategy in working out a proof.

Vocabularyuniquely determined

auxiliary fi gure

non-Euclidean geometries

obtuse triangle

right triangle

acute triangle

Some Examples of UniquenessThe adjective unique means “exactly one.” When exactly one thing satisfi es some given conditions, we say the thing is uniquely determined. For instance, your address uniquely determines which building you live in. In algebra, the given condition 4x + 7 = 31 uniquely determines the value of x.

Example 1Given a segment

___ AB , which of these things are uniquely determined?

If they are not, why not?

a. midpoint of ___

AB b. bisector of

___ AB

c. perpendicular bisector of ___

AB

Solution

a. Does a segment have exactly one midpoint? Yes

b. Does a segment have exactly one bisector? No. There can be many

lines, segments, or rays that pass through the midpoint, and

each is a bisector because of the defi nition of “bisector.”

c. Does a segment have exactly one perpendicular bisector? Yes

It is possible to prove that certain fi gures are unique. For example, in Lesson 5-5, the circle through three noncollinear points A, B, and C is unique because (1) the perpendicular bisector of any segment contains all points equidistant from the endpoints of the segment, (2) the two perpendicular bisectors of

___ AB and

___ BC

intersect at a unique point, and (3) there is a unique circle with a particular center and radius. So we can assert the following theorem.

In the fi gure below, −−

AD � −−

BC , −−

AB � −−

DC , and m∠A = 45. Find:

a. m∠D.

b. m∠B.

c. m∠C.

Mental Math

READING MATH

The prefi x uni (from the Latin unus), meaning “one,” starts many English words. These include unicycle, uniform, unilateral, union, and universal.

A

C

D

B

O

A

CB

135

135

45

Lesson

5-6

282 Chapter 5

GOALLearn the idea of justifying auxiliary fi gures and use them to deduce that (1) there is exactly one parallel to a given line through a point not on it and (2) the sum of the measures of the angles of a triangle is 180º.

SPUR ObjectivesD Use the Triangle-Sum Theorem to determine angle measures.

I Tell whether auxiliary fi gures are uniquely determined.

K Draw fi gures and auxiliary fi gures to aid proofs.

Materials/Resources• Lesson Masters 5-6A and 5-6B• Resource Masters 82 and 83• Quiz 2

HOMEWORKSuggestions for Assignment• Questions 1–27• Question 28 (extra credit)• Reading Lesson 5-7• Covering the Ideas 5-7

Local Standards

Suppose you wanted to prove the following statement: If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180º, then the lines will intersect on that same side of the transversal. Draw a fi gure and state the Given and To Prove for your fi gure.

Warm-Up1 Warm-Up1Background

The importance of the idea of auxiliary fi gures is demonstrated by how little could be proved without them. Almost any nonobvious theorem requires at least one auxiliary line. We used auxiliary fi gures in the proofs of the Two-Refl ection Theorems in Lessons 4-4 and 4-5. The idea of auxiliary fi gures is not new; only their justifi cation is new.

How many parallels are there? Because perpendiculars are unique, does the same apply to parallels? One of the greatest discoveries in mathematics is that one can

assume nonuniqueness of parallels and still have consistent geometry. The Russian mathematician Nikolai Ivanovich Lobachevski and the Hungarian mathematician János Bolyai independently discovered this in the late 1820s.

Lobachevski and Bolyai wanted to prove Euclid’s fi fth postulate from the other four. They began with indirect reasoning, assuming a negation of Playfair’s Postulate. Instead of arriving at a contradiction, they were able to deduce many theorems.

Answers vary. Sample (on next page):

SMP_TEGEO_C05L06_282-287.indd 282 4/11/08 10:50:53 AM

18 Call 1-800-648-2970

Auxiliary Figures and Uniqueness 283

Lesson 5-6

Unique Circle Theorem

There is exactly one circle (a unique circle) through three given noncolinear points.

QY

What Are Auxiliary Figures?A segment, line, or other fi gure that is added to a diagram is called an auxiliary figure. The word auxiliary means “assisting” or “giving help.”

When an auxiliary fi gure is not uniquely determined, then there are two possibilities: (1) There may be more than one fi gure satisfying the conditions. This is the case with bisectors of segments: (2) There may be no fi gure satisfying the given conditions.

Example 2In quadrilateral ABCD, a student wished to draw as an auxiliary segment, the diagonal

___ AC that bisects ∠A. Is this always possible? Why or why not?

Solution It is not possible to do this in every quadrilateral.

Draw a quadrilateral ABCD. Diagonal

___

AC is uniquely determined

because points A and C determine a line. However,

___

AC does not

have to be the bisector of ∠DAB, as the fi gure shows.

Activity1. Draw a quadrilateral ABCD, different from the one in Example 2, in which

the diagonal __

AC does not lie on the bisector of ∠A.

2. Draw a quadrilateral EFGH in which the diagonal __

FH does appear to lie on the bisector of ∠F.

How Many Parallels Are There?As Example 2 demonstrates, uniqueness is not always obvious. The following theorem addresses the question of how many lines can be drawn that are parallel to a given line � and that pass through some given point not on �. In the proof, two auxiliary lines are drawn. We do this to create alternate interior angles that are congruent.

QY

Draw two points A and B. Show that the circle containing A and B is not uniquely determined.

BA

CD

B

A

CD

E

H

F

G

Activity

1.

2.

5-6

Lesson 5-6 283

Other mathematicians had followed the same approach but thought that they had not reasoned long enough to get a contradiction. The brilliance of Lobachevski and Bolyai was to realize that there was no such contradiction to be found. Other mathematicians later verifi ed that Lobachevski and Bolyai were correct by describing planes, an undefi ned term, that did not satisfy Playfair’s Parallel Postulate.

Today’s geometry texts tend to be idiosyncratic; two texts seldom have exactly the same set of postulates. The set in this book differs from the set used in others. We substitute the Corresponding Angle Postulate for Euclid’s Parallel Postulate and Playfair’s Parallel Postulate (see page 284). This does not change the set of possible propositions (postulates, theorems, and defi nitions) that are in the geometry. We are still in Euclidean geometry.

Additional ExamplesExample 1 Tell if each thing is uniquely determined. If not, why not?

a. Given ∠DEF, bisector of ∠DEF

b. Given point A on line m, point X on m a given distance from A

c. Given line n and point C on n, the perpendicular to n through C

Example 2 In �ABC below, a student wished to draw as an auxiliary line the perpendicular bisector of

___ AC that

passes through the vertex of ∠B. Is this always possible? Why or why not?

C

DA

EB

Teaching2 Teaching2

No; there are two points X on m the same distance from A, one on each side of A.

It is not possible to do this in every triangle. Draw a �ABC. The perpendicular bisector of

−− AC is

uniquely determined by −−

AC . However, � ⎯ � DE does not have to pass through the vertex of ∠B, as the fi gure shows.

yes

yes

C

AD

F

H

85°

B

Given: m∠BHF + m∠DFH < 180º; To Prove: � ⎯ � AB and � ⎯ � CD intersect on the same side of � ⎯ � HF as points B and D.

SMP_TEGEO_C05L06_282-287.indd 283 4/28/08 12:16:08 PM


LessonAuxiliary Figures and Uniqueness

Chapter 5

5-6

BIG IDEA Introducing a line or part of a line into a fi gure can be a helpful strategy in working out a proof.

Vocabularyuniquely determined

auxiliary fi gure

non-Euclidean geometries

obtuse triangle

right triangle

acute triangle

Some Examples of UniquenessThe adjective unique means “exactly one.” When exactly one thing satisfi es some given conditions, we say the thing is uniquely determined. For instance, your address uniquely determines which building you live in. In algebra, the given condition 4x + 7 = 31 uniquely determines the value of x.

Example 1Given a segment

___ AB , which of these things are uniquely determined?

If they are not, why not?

a. midpoint of ___

AB b. bisector of

___ AB

c. perpendicular bisector of ___

AB

Solution

a. Does a segment have exactly one midpoint? Yes

b. Does a segment have exactly one bisector? No. There can be many

lines, segments, or rays that pass through the midpoint, and

each is a bisector because of the defi nition of “bisector.”

c. Does a segment have exactly one perpendicular bisector? Yes

It is possible to prove that certain fi gures are unique. For example, in Lesson 5-5, the circle through three noncollinear points A, B, and C is unique because (1) the perpendicular bisector of any segment contains all points equidistant from the endpoints of the segment, (2) the two perpendicular bisectors of

___ AB and

___ BC

intersect at a unique point, and (3) there is a unique circle with a particular center and radius. So we can assert the following theorem.

In the fi gure below, −−

AD � −−

BC , −−

AB � −−

DC , and m∠A = 45. Find:

a. m∠D.

b. m∠B.

c. m∠C.

Mental Math

READING MATH

The prefi x uni (from the Latin unus), meaning “one,” starts many English words. These include unicycle, uniform, unilateral, union, and universal.

A

C

D

B

O

A

CB

135

135

45

Lesson

5-6

282 Chapter 5

GOALLearn the idea of justifying auxiliary fi gures and use them to deduce that (1) there is exactly one parallel to a given line through a point not on it and (2) the sum of the measures of the angles of a triangle is 180º.

SPUR ObjectivesD Use the Triangle-Sum Theorem to determine angle measures.

I Tell whether auxiliary fi gures are uniquely determined.

K Draw fi gures and auxiliary fi gures to aid proofs.

Materials/Resources• Lesson Masters 5-6A and 5-6B• Resource Masters 82 and 83• Quiz 2

HOMEWORKSuggestions for Assignment• Questions 1–27• Question 28 (extra credit)• Reading Lesson 5-7• Covering the Ideas 5-7

Local Standards

Suppose you wanted to prove the following statement: If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180º, then the lines will intersect on that same side of the transversal. Draw a fi gure and state the Given and To Prove for your fi gure.

Warm-Up1 Warm-Up1Background

The importance of the idea of auxiliary fi gures is demonstrated by how little could be proved without them. Almost any nonobvious theorem requires at least one auxiliary line. We used auxiliary fi gures in the proofs of the Two-Refl ection Theorems in Lessons 4-4 and 4-5. The idea of auxiliary fi gures is not new; only their justifi cation is new.

How many parallels are there? Because perpendiculars are unique, does the same apply to parallels? One of the greatest discoveries in mathematics is that one can

assume nonuniqueness of parallels and still have consistent geometry. The Russian mathematician Nikolai Ivanovich Lobachevski and the Hungarian mathematician János Bolyai independently discovered this in the late 1820s.

Lobachevski and Bolyai wanted to prove Euclid’s fi fth postulate from the other four. They began with indirect reasoning, assuming a negation of Playfair’s Postulate. Instead of arriving at a contradiction, they were able to deduce many theorems.

Answers vary. Sample (on next page):




Chapter 5

Uniqueness of Parallels Theorem

Through a point not on a line, there is exactly one line parallel to the given line.

Given Point P not on line �,points R and Q on line �.

Prove There is exactly one line parallel to � through P.

Proof Draw � � PQ . By the Point-Line-Plane Postulate, �

� PQ is uniquely

determined. We label ∠PQR as ∠1. (See the fi gure at the right.) Now draw �

� PA so that A is on the other side of �

� PQ from R and

m∠APQ = m∠1. � � PA is unique because of the Unique Angle

Assumption in the Angle Measure Postulate. � � PA � �

� RQ by the

Alternate Interior Angles Theorem. So there is at least one line parallel to � through P.

Can there be another parallel? By the Alternate Interior Angles Theorem, m∠QPA for every line parallel to � through P is the same. Since in a given side of a line there is only one angle with this measure (Angle Measure Postulate), there cannot be more than one parallel. Thus, �

� PA is unique and there is exactly one line

parallel to � through P.

Playfair’s Parallel PostulateThe Uniqueness of Parallels Theorem is important in the history of mathematics. It ultimately changed the entire nature of mathematics. In Euclid’s Elements, the fi fth and fi nal geometric postulate is: If two lines are cut by a transversal, and the measures of the same-side interior angles sum to less than 180º, then the lines will intersect on that side of the transversal. This postulate bothered mathematicians, who felt that such a complicated statement should not be assumed true. For 2000 years they tried to prove the fi fth postulate from Euclid’s other postulates.

After centuries of being unable to prove Euclid’s fi fth postulate, some mathematicians substituted simpler statements for it. The uniqueness of parallels statement above was fi rst suggested by the Greek mathematician Proclus about 450 CE, but it is known as Playfair’s Parallel Postulate because it was used by the Scottish mathematician John Playfair in 1795. We were able to prove it as a theorem in this lesson because we assumed the Corresponding Angles Postulate in Lesson 3-6. With that postulate, we proved the fi rst part of the Parallel Lines Theorem and the Alternate Interior Angles Theorem.

Q

R

P �

Q

R

P

A

�

1

Mathematician and geologist John Playfair

5-6

284 Chapter 5

Notes on the LessonPlayfair’s Parallel Posulate The following statements are logically equivalent; that is, from any statement, all the others can be deduced.

• If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180º, then the lines will intersect on that side of the transversal. (Euclid’s Parallel Postulate)

• Through a point not on a line, there is exactly one line parallel to the given line. (Playfair’s Parallel Postulate)

• If two parallel lines are cut by a transversal, then corresponding angles have the same measure. (Corresponding Angles Postulate)

• The sum of the measures of the angles of a triangle is 180º. (Triangle-Sum Theorem)

• If � � m and m � n, then � � n. (Transitivity of Parallelism Theorem)

The above statements, and many others not listed here, have been used by mathematicians as a way of deducing the properties of parallel lines in Euclidean geometry. Any of the postulates essentially plays the role of guaranteeing that the plane is not bent.

Uniquely determined fi gures You might give students a list of those fi gures that are uniquely determined and the reason that they are. Through Lesson 5-5, there have been at least 11 uniquely determined fi gures.

• given two points, the segment joining them (Point-Line-Plane Postulate part a)

• given a ray and a distance x, the point on the ray at that distance from the endpoint (Point-Line-Plane Postulate part a and Distance Postulate part a)

• given a segment, its midpoint (defi nition of midpoint)

• given a ray and a measure x between 0º and 180º, an angle with the ray as one side on a given half-plane of the ray and with measure x (Angle Measure Postulate)

• given an angle, its bisector (defi nition of angle bisector and Angle Measure Postulate)

• given a line and a point, the perpendicular to that line through that point (uniqueness of angle, defi nition of perpendicular, and Linear Pair Theorem)

ENGLISH LEARNERS


Have students use a dictionary to fi nd the meanings of obtuse, right, and acute and try to relate the English meanings (not sharp for obtuse and sharp for acute) with these types of triangles. Be sure that students understand that triangles are classifi ed by their largest angle. Then have them draw examples of each type of triangle.


Have students briefl y research a non-Euclidean geometry and write a report (up to two pages in length) to be shared with the class. They can fi nd information on the Internet. Have them discuss how the geometry they have researched differs from Euclidean geometry.


20 Call 1-800-648-2970


Lesson 5-6

By the nineteenth century, other mathematicians had substituted different statements for Playfair’s Parallel Postulate. When they assumed there are no parallels to a line through a point not on it, they were able to develop a spherical geometry that could apply to the surface of the Earth. When they assumed there is more than one parallel to a line through a point not on it, they developed types of geometries for other surfaces. The most notable of these is called hyperbolic geometry. All of these geometries are called non-Euclidean geometries. Non-Euclidean geometries are important in physics in the theory of relativity.

These mathematicians greatly infl uenced all later mathematics with their work. For the fi rst time, postulates were viewed as statements assumed true instead of statements defi nitely true. With this point of view, mathematicians experimented with a variety of algebras and types of geometries formed by modifying or changing postulates. A useful algebra, with some postulates different from those you have studied, is applied in logic and in the operation of computers.

Proving the Triangle-Sum TheoremIn previous courses you learned that the sum of the measures of the three angles in any triangle is 180º. A nice consequence of the Uniqueness of Parallels Theorem is that it enables a short proof of the Triangle-Sum Theorem.

Triangle-Sum Theorem

The sum of the measures of the angles of any triangle is 180º.

Given �ABC

Prove m∠A + m∠B + m∠C = 180

Proof Draw auxiliary line � � BD with �

� BD � �

� AC . �

� BD exists because of

the Uniqueness of Parallels Theorem. Pick a point E on � � BD

such that B is on ___

ED . Label angles 1, 2, and 3 as shown.

Notice that ∠A and ∠1 are alternate interior angles, as are ∠C and ∠3. Because �

� BD � �

� AC , these alternate interior

angles must be congruent by the Parallel Lines Theorem. Thus, m∠A = m∠1 and m∠C = m∠3. We also know m∠EBD = 180 because it is a straight angle. By the Angle Addition Property, m∠1 + m∠2 + m∠3 = m∠EBD. Now, by substitution, m∠A + m∠B + m∠C = 180.

This argument proves the theorem you have used for years.

123

D

A C

BE

5-6

Lesson 5-6 285

• given a segment, its perpendicular bisector (defi nition of perpendicular bisector, Distance Postulate, and Angle Measure Postulate)

• given a point and a line, the refl ection image of that point (Refl ection Postulate part a)

• given a point P and a transformation T, the image point T(P) (defi nition of transformation)

• given a point and a line, the perpendicular to that line from that point (Refl ection Postulate)

• given three noncollinear points, the circle through them (proved in Example 1, Lesson 5-5)

You may wish to emphasize the following facts: (1) The theorems in a mathematical system depend on the postulates chosen. (2) The applicability of a mathematical system depends on the postulates chosen. (3) Changing the postulates may change the set of possible theorems that can be proved and the applicability of the theorems. (4) Changing the postulates usually does change which theorems are easier to prove than others.


Have students justify these triangle relationships. You might have them draw and measure diagrams to support their fi ndings.

1. In an equiangular triangle, each angle measures 60º.

2. In a right isosceles triangle, each congruent angle measures 45º.

3. In an obtuse isosceles triangle, each congruent angle measures less than 45º.

4. In an acute isosceles triangle, each congruent angle has a measure that is greater than 45º but less than 90º.

Answers vary. Check students’ work.

Additional Answers

5. The midpoints of segments are uniquely

determined, and given two points, the

segment connecting these two points is

also uniquely determined.

6. The Uniqueness of Parallels Theorem is

also called Playfair’s Parallel Postulate

because the Scottish mathematician

John Playfair used it in 1795.

7. Two auxiliary lines are used, one parallel

to the fi rst line passing through the point

P, and a transversal to the two parallel

lines and passing through P.

11. A right angle has measure 90 and an

obtuse angle measures greater than 90.

Their measures sum to over 180, which

is impossible in a triangle.



Chapter 5


Through a point not on a line, there is exactly one line parallel to the given line.

Given Point P not on line �,points R and Q on line �.

Prove There is exactly one line parallel to � through P.

Proof Draw � � PQ . By the Point-Line-Plane Postulate, �

� PQ is uniquely

determined. We label ∠PQR as ∠1. (See the fi gure at the right.) Now draw �

� PA so that A is on the other side of �

� PQ from R and

m∠APQ = m∠1. � � PA is unique because of the Unique Angle

Assumption in the Angle Measure Postulate. � � PA � �

� RQ by the

Alternate Interior Angles Theorem. So there is at least one line parallel to � through P.

Can there be another parallel? By the Alternate Interior Angles Theorem, m∠QPA for every line parallel to � through P is the same. Since in a given side of a line there is only one angle with this measure (Angle Measure Postulate), there cannot be more than one parallel. Thus, �

� PA is unique and there is exactly one line

parallel to � through P.

Playfair’s Parallel PostulateThe Uniqueness of Parallels Theorem is important in the history of mathematics. It ultimately changed the entire nature of mathematics. In Euclid’s Elements, the fi fth and fi nal geometric postulate is: If two lines are cut by a transversal, and the measures of the same-side interior angles sum to less than 180º, then the lines will intersect on that side of the transversal. This postulate bothered mathematicians, who felt that such a complicated statement should not be assumed true. For 2000 years they tried to prove the fi fth postulate from Euclid’s other postulates.

After centuries of being unable to prove Euclid’s fi fth postulate, some mathematicians substituted simpler statements for it. The uniqueness of parallels statement above was fi rst suggested by the Greek mathematician Proclus about 450 CE, but it is known as Playfair’s Parallel Postulate because it was used by the Scottish mathematician John Playfair in 1795. We were able to prove it as a theorem in this lesson because we assumed the Corresponding Angles Postulate in Lesson 3-6. With that postulate, we proved the fi rst part of the Parallel Lines Theorem and the Alternate Interior Angles Theorem.

Q

R

P �

Q

R

P

A

�

1

Mathematician and geologist John Playfair

5-6

284 Chapter 5

Notes on the LessonPlayfair’s Parallel Posulate The following statements are logically equivalent; that is, from any statement, all the others can be deduced.

• If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180º, then the lines will intersect on that side of the transversal. (Euclid’s Parallel Postulate)

• Through a point not on a line, there is exactly one line parallel to the given line. (Playfair’s Parallel Postulate)

• If two parallel lines are cut by a transversal, then corresponding angles have the same measure. (Corresponding Angles Postulate)

• The sum of the measures of the angles of a triangle is 180º. (Triangle-Sum Theorem)

• If � � m and m � n, then � � n. (Transitivity of Parallelism Theorem)

The above statements, and many others not listed here, have been used by mathematicians as a way of deducing the properties of parallel lines in Euclidean geometry. Any of the postulates essentially plays the role of guaranteeing that the plane is not bent.

Uniquely determined fi gures You might give students a list of those fi gures that are uniquely determined and the reason that they are. Through Lesson 5-5, there have been at least 11 uniquely determined fi gures.

• given two points, the segment joining them (Point-Line-Plane Postulate part a)

• given a ray and a distance x, the point on the ray at that distance from the endpoint (Point-Line-Plane Postulate part a and Distance Postulate part a)

• given a segment, its midpoint (defi nition of midpoint)

• given a ray and a measure x between 0º and 180º, an angle with the ray as one side on a given half-plane of the ray and with measure x (Angle Measure Postulate)

• given an angle, its bisector (defi nition of angle bisector and Angle Measure Postulate)

• given a line and a point, the perpendicular to that line through that point (uniqueness of angle, defi nition of perpendicular, and Linear Pair Theorem)

ENGLISH LEARNERS


Have students use a dictionary to fi nd the meanings of obtuse, right, and acute and try to relate the English meanings (not sharp for obtuse and sharp for acute) with these types of triangles. Be sure that students understand that triangles are classifi ed by their largest angle. Then have them draw examples of each type of triangle.


Have students briefl y research a non-Euclidean geometry and write a report (up to two pages in length) to be shared with the class. They can fi nd information on the Internet. Have them discuss how the geometry they have researched differs from Euclidean geometry.




Chapter 5

Because the sum of the measures of the angles of a triangle is fi xed at 180, a triangle cannot have more than one angle that is right or obtuse. For this reason we can classify triangles by their largest angle. An obtuse triangle is a triangle with an obtuse angle. A right triangle is a triangle with a right angle. An acute triangle is a triangle with all three angles acute.


In 1−4, tell whether the fi gure is or is not uniquely determined in Euclidean geometry and explain why.

1. line parallel to a given line 2. line parallel to two given lines 3. line perpendicular to a given line and through a point not on the

given line 4. point equidistant from the endpoints of a given segment

5. Explain why the segment connecting the midpoints of sides ___

AB and

___ AC of �ABC is uniquely determined.

In 6–8, refer to the Uniqueness of Parallels Theorem.

6. Why is this theorem also called Playfair’s Parallel Postulate? 7. What auxiliary fi gures are used in its proof? 8. Give the justifi cation for each conclusion. a. There is a unique line determined by points P and Q. b. There is a unique line containing

� PA so that m∠APQ = m∠1

and ∠APQ and ∠1 are alternate interior angles.

9. What postulate in this book substitutes for Playfair’s Parallel Postulate and Euclid’s fi fth postulate?

10. Give an example of a non-Euclidean geometry.

11. Why can’t a triangle have one right angle and one obtuse angle?

12. Redraw �ABC from the proof of the Triangle-Sum Theorem. Suppose you were going to redo the proof using an auxiliary line drawn through A instead of an auxiliary line drawn through B.

a. To what line would the auxiliary line through A be parallel? b. Why would the auxiliary line be unique? c. Draw the diagram of �ABC and the auxiliary line through A.

APPLYING THE MATHEMATICS

In 13 and 14, tell whether the number described is or is not uniquely determined and why.

13. solution to x2 = 25 14. measure of a right angle

1. No, in Euclidean geometry, there are infi nitely many distinct pairs of parallel lines.

2. No. First of all, such a line may not exist because the two other lines may intersect. If the two other lines are parallel, then there are still infi nitely many lines that are also parallel to these lines in Euclidean geometry.

3. Yes, in Euclidean geometry, a line is uniquely determined by a point and slope.

4. No, any point on the perpendicular bisector of that segment is equidistant from the two endpoints.

8a. Unique Line Assumption of the Point-Line-Plane Postulate

8b. Unique Angle Assumption of the Angle Measure Postulate

10. Answers vary. Sample: spherical or hyperbolic geometry

6–7. See margin.

Corresponding Angles Postulate

11–12. See margin.


See margin.

5-6

286 Chapter 5

Recommended Assignment• Questions 1–27

• Question 28 (extra credit)



Notes on the QuestionsFor most classes, review Questions 1–10 to insure that the basic ideas of the lesson are clear to students.

Questions 1–4 You might ask students to suggest other examples if you have not given the list mentioned in the Notes on the Lesson.

Question 11 There are many variant questions that have the same answer. Why can’t a triangle have two right angles? Two obtuse angles? Three obtuse angles? In all cases, the measures of the angles would add to greater than 180º, thus contradicting the Triangle-Sum Theorem.


Geometry 241

Copyright ©

Wright G

roup/McG

raw-H

ill

5-1B Lesson Master Questions on SPUR ObjectivesSee Student Edition pages 302–305 for objectives.

5-6A

SKILLS Objective D

1. Two of the angles of a triangle measure 84 and 71. Find the measure of the third angle.

2. The measure of one angle of an isosceles triangle is three times the measure of each of the other two angles. Find the measure of the largest angle.

3. Refer to the fi gure below 4. Refer to the fi gure belowto fi nd the value of x. to fi nd the value of a.

(3x + 13)˚

(3x - 5)˚(2x + 4)˚

(4a + 7)˚

88˚

(a + 12)˚

PROPERTIES Objective I

In 5−7, tell whether the fi gure described is uniquely determined.

5. A line perpendicular to a given segment ___

AB .

6. Given a point A on a circle C, a diameter of C that contains A.

7. The longest diagonal of a given pentagon.

REPRESENTATIONS Objective K

8. In �SCA (not shown), a student wishes to draw an auxiliary �� SL to bisect

∠CSA such that �� SL is ⊥ to

___ CA . Is this possible? Why or why not? Draw a

picture to support your explanation.

9. Ali is trying to prove a theorem about the perpendicular bisectors G

HF

of the sides of an obtuse triangle. Draw the perpendicular bisectors of sides

___ FG and

___ GH and their intersection. Mark the

fi gure to show perpendicular lines and congruent segments.

25

108

13.821

not uniquely determineduniquely determined

not uniquely determined

Answers vary. Sample: This is possible only if

___ CS �

___ AS . A

L

C

S

Additional Answers

12a. ��BC

12b. Uniqueness of Parallels Theorem

12c.

A

B

C

Additional Answers

13. Not uniquely determined because both 5 and

–5 are solutions.

14. Uniquely determined because by defi nition, a

right angle has measure 90.

16. Uniquely determined. Answers vary. Sample: A

B

C

D

17. Uniquely determined. Answers vary. Sample:

AB

C

D


A

MB

C

D


22 Call 1-800-648-2970


Lesson 5-6

In 15−19, given a quadrilateral ABCD, tell whether the auxiliary fi gure is uniquely determined. If so, make a drawing of this auxiliary fi gure; if not explain why not. You may fi nd a DGS helpful.

15. line perpendicular to side ____

AD 16. intersection point of the diagonals

___ AC and

____ BD

17. angle bisector of ∠ACD 18. point M on side

____ AD such that AM = MD

19. point of intersection N of the perpendicular bisectors of sides

___ BC and

____ AD

20. Use the fi gure at the right. Given m � n, fi nd x.

21. Use the fi gure for Question 20. Replace 45 by a, 150 by b, and fi nd a formula for x in terms of a and b. You might fi nd a DGS helpful.

22. Find an equation for the line that is parallel to the line 3x + 4y = 11, and contains the point (8, 0).

23. Natane was supposed to prove a theorem involving the fi gure at the right. She decided that she needed an auxiliary line through R that was parallel to side

___ PQ . What

justifi cation could she give for this step?

REVIEW

24. In the fi gure at the right AE = CE and m∠CED = 90. Prove that �BAD � �BCD. (Lesson 5-5)

25. Find the center of rotation and magnitude for rx-axis � ry-axis. (Lesson 4-5)

26. Consider the lines y = –4.5, y = 13, y = 100, y = 0, y = –28. The composite of refl ections over which two of these lines gives a translation with the greatest magnitude? (Lesson 4-4)

27. Suppose a triangle has sides of length 3, z, and z + 2, and z is an integer. Is z uniquely determined? (Lesson 1-7)

EXPLORATION

28. Use a DGS to complete the following construction:Step 1 On a clear DGS screen, construct a triangle �ABC.

Step 2 Construct the line parallel to __

AB through C.

Step 3 Construct the line parallel to ___

BC through A.

Step 4 Construct the line parallel to __

AC through B.

Make and try to prove conjectures about the fi gure formed when all of these lines are constructed.

m

n

x°

45°

150°

A

B

C

D

E

S

R

P Q

QY ANSWER

A B

26. y = 100 and y = –28


15. Not uniquely determined because there are infi nitely many lines perpendicular to a given line.

75

x = 180 + a – b

y = – 3 __ 4 x + 6


See margin.

The origin is the center; the magnitude is 180.

no

See margin.

5-6

Lesson 5-6 287

Geometry 243

Copyright ©

Wright G

roup/McG

raw-H

ill

5-1B page 2

11. line parallel to ___

ZY through W

YX

Z

W

12. line through H and K perpendicular to m

H

Km

13. perpendicular bisector of ___

RT

E

T

R

14. circle containing the vertices of �SEH

H

E

S


15. A student wished to draw, as an auxiliary fi gure, line u parallel to two given lines e and r. Explain if this is possible. Use a diagram if you wish.

16. Describe the construction of the auxiliary line used to prove the Triangle Sum Theorem.

5-6B

unique

not unique; no figure

unique

unique

u u

or

rre e

Answers vary. Sample: Not possible; u � e or u � r but not both, unless e � r.

Draw a line through the vertex of one angle that is parallel to the opposite side by constructing an alternate interior angle congruent to one of the other angles.

242 Geometry

Cop

yrig

ht ©

Wrig

ht G

roup

/McG

raw

-Hill

5-1A Questions on SPUR ObjectivesSee Student Edition pages 302–305 for objectives.

Lesson Master5-6B

SKILLS Objective D

1. What is the sum of the measures of the angles of a scalene triangle?

In 2–4, refer to the fi gure at the right.

5a

2a

3a

Q

T P R

2. a =

3. m∠QPR =

4. Why is �QPT a right triangle?

5. The measures of the angles of a triangle are in the ratio 8:6:2. Find the measure of the largest angle.

6. In triangle ABC, m∠A = 80. The measure of ∠B is 15 more than 9 times the measure of ∠C. Find

a. m∠B. b. m∠C.


In 7–14, tell whether the fi gure described is unique. If not, tell whether there is more than one fi gure or no fi gure satisfying the description.

7. midpoint of ___

UV 8. bisector of ____

MN

V

U

N

G

M

9. diagonal AC bisecting ∠A 10. point R between P and Q

B

C

D

E

A

QP

18

180

126

90

91.5 8.5

unique not unique; more than one figure

not unique; no figure

not unique; more than one figure

The measure of ∠T is 90.

Notes on the QuestionsQuestion 22 This is a standard algebra problem in which the fact that there is always exactly one such line provides a coordinate proof of Playfair’s Parallel Postulate.

Ongoing AssessmentHave students trace �ABC from the proof of the Triangle-Sum Theorem and draw the auxiliary line through A as specifi ed in Question 12. Then have them write the proof of the theorem using this diagram.

Administer Quiz 2 (or your own quiz) after students have completed this lesson.

Project UpdateProject 1, Triangles on Curved Surfaces, on page 296, relates to the content of this lesson.

Wrap-Up4 Wrap-Up4

Students should write a proof of the Triangle-Sum Theorem using an auxiliary line different from the line in the text proof.

Additional Answers


A

N

B

C

D

24. Answers vary. Sample: Since AE = EC is

the given, we know that C is the refl ection

image of A over −−

DB by the defi nition of

refl ections. Therefore, AD = DC and AB = BC

by the defi nition of refl ections. Thus, by the

defi nition of congruence, �BAD � �BCD.

28. Answers vary. Sample: Conjecture: each of

the four smaller triangles is congruent. Proof:

By construction, the vertices of the larger

triangle are each refl ections of the points

A, B, and C with respect to BC, AC, and AB,

respectively. Because angles and distances

are preserved under refl ections, the new

triangles are all congruent to the original.



Chapter 5

Because the sum of the measures of the angles of a triangle is fi xed at 180, a triangle cannot have more than one angle that is right or obtuse. For this reason we can classify triangles by their largest angle. An obtuse triangle is a triangle with an obtuse angle. A right triangle is a triangle with a right angle. An acute triangle is a triangle with all three angles acute.


In 1−4, tell whether the fi gure is or is not uniquely determined in Euclidean geometry and explain why.

1. line parallel to a given line 2. line parallel to two given lines 3. line perpendicular to a given line and through a point not on the

given line 4. point equidistant from the endpoints of a given segment

5. Explain why the segment connecting the midpoints of sides ___

AB and

___ AC of �ABC is uniquely determined.

In 6–8, refer to the Uniqueness of Parallels Theorem.

6. Why is this theorem also called Playfair’s Parallel Postulate? 7. What auxiliary fi gures are used in its proof? 8. Give the justifi cation for each conclusion. a. There is a unique line determined by points P and Q. b. There is a unique line containing

� PA so that m∠APQ = m∠1

and ∠APQ and ∠1 are alternate interior angles.

9. What postulate in this book substitutes for Playfair’s Parallel Postulate and Euclid’s fi fth postulate?

10. Give an example of a non-Euclidean geometry.

11. Why can’t a triangle have one right angle and one obtuse angle?

12. Redraw �ABC from the proof of the Triangle-Sum Theorem. Suppose you were going to redo the proof using an auxiliary line drawn through A instead of an auxiliary line drawn through B.

a. To what line would the auxiliary line through A be parallel? b. Why would the auxiliary line be unique? c. Draw the diagram of �ABC and the auxiliary line through A.

APPLYING THE MATHEMATICS

In 13 and 14, tell whether the number described is or is not uniquely determined and why.

13. solution to x2 = 25 14. measure of a right angle

1. No, in Euclidean geometry, there are infi nitely many distinct pairs of parallel lines.

2. No. First of all, such a line may not exist because the two other lines may intersect. If the two other lines are parallel, then there are still infi nitely many lines that are also parallel to these lines in Euclidean geometry.

3. Yes, in Euclidean geometry, a line is uniquely determined by a point and slope.

4. No, any point on the perpendicular bisector of that segment is equidistant from the two endpoints.

8a. Unique Line Assumption of the Point-Line-Plane Postulate

8b. Unique Angle Assumption of the Angle Measure Postulate

10. Answers vary. Sample: spherical or hyperbolic geometry

6–7. See margin.

Corresponding Angles Postulate



See margin.

5-6

286 Chapter 5

Recommended Assignment• Questions 1–27

• Question 28 (extra credit)



Notes on the QuestionsFor most classes, review Questions 1–10 to insure that the basic ideas of the lesson are clear to students.

Questions 1–4 You might ask students to suggest other examples if you have not given the list mentioned in the Notes on the Lesson.

Question 11 There are many variant questions that have the same answer. Why can’t a triangle have two right angles? Two obtuse angles? Three obtuse angles? In all cases, the measures of the angles would add to greater than 180º, thus contradicting the Triangle-Sum Theorem.


Geometry 241

Copyright ©

Wright G

roup/McG

raw-H

ill

5-1B Lesson Master Questions on SPUR ObjectivesSee Student Edition pages 302–305 for objectives.

5-6A

SKILLS Objective D

1. Two of the angles of a triangle measure 84 and 71. Find the measure of the third angle.

2. The measure of one angle of an isosceles triangle is three times the measure of each of the other two angles. Find the measure of the largest angle.

3. Refer to the fi gure below 4. Refer to the fi gure belowto fi nd the value of x. to fi nd the value of a.

(3x + 13)˚

(3x - 5)˚(2x + 4)˚

(4a + 7)˚

88˚

(a + 12)˚


In 5−7, tell whether the fi gure described is uniquely determined.

5. A line perpendicular to a given segment ___

AB .

6. Given a point A on a circle C, a diameter of C that contains A.

7. The longest diagonal of a given pentagon.


8. In �SCA (not shown), a student wishes to draw an auxiliary �� SL to bisect

∠CSA such that �� SL is ⊥ to

___ CA . Is this possible? Why or why not? Draw a

picture to support your explanation.

9. Ali is trying to prove a theorem about the perpendicular bisectors G

HF

of the sides of an obtuse triangle. Draw the perpendicular bisectors of sides

___ FG and

___ GH and their intersection. Mark the

fi gure to show perpendicular lines and congruent segments.

25

108

13.821

not uniquely determineduniquely determined

not uniquely determined

Answers vary. Sample: This is possible only if

___ CS �

___ AS . A

L

C

S

Additional Answers

12a. ��BC

12b. Uniqueness of Parallels Theorem

12c.

A

B

C

Additional Answers

13. Not uniquely determined because both 5 and

–5 are solutions.

14. Uniquely determined because by defi nition, a

right angle has measure 90.

16. Uniquely determined. Answers vary. Sample: A

B

C

D


AB

C

D


A

MB

C

D



Chapter

Projects5

Chapter 5

1 Triangles on Curved SurfacesA special kind of non-Euclidean

geometry is called spherical geometry. In spherical geometry, the points are all on a sphere. To investigate this geometry you will need a sphere that you can write on, a washable marker, 3 or more pieces of string, and a protractor. a. Choose two points on the sphere. Stretch a

piece of string from one point to the other so that the string is as short as possible. This shortest path between two points can be thought of as a “line segment” in spherical geometry. A “line” through two points in spherical geometry is the path on the sphere that contains the shortest path from one point to another. If a line in spherical geometry is drawn, what is the result?

b. Create three points on the sphere. Create a “triangle” and measure its angles to the best of your ability. Do this for several different triangles. Make a conjecture about the sum of the measures of a triangle in spherical geometry.

2 Congruence and GeneticsHuman DNA is made up of four basic

building blocks. Any two blocks of the same kind should be congruent. Find out what these blocks are. Why do you think it is important that they be congruent? Make a model of a section of human DNA, illustrating the different building blocks.

3 Congruence and LiteracyJohannes Gutenberg is credited with

inventing the fi rst printing press—one of the most important inventions ever. The printing press produces many congruent copies of the same image. Find out how the printing press works, and prepare a presentation on this. Include some modern developments of this kind of machine, and how the methods of producing congruent images have changed over time.


296 Chapter 5

5Chapter

The projects relate to the content of the lessons of this chapter as follows:

Project Lesson(s)

1 5-6

2 5-1

3 5-1

4 5-4

5 5-5

6 5-7

1 Triangles on Curved Surfaces

Students may fi nd it interesting to check their ideas about angle measures on a sphere by consulting mathematics, geography, and cartography books. Spherical trigonometry was once a fairly common topic in textbooks, though it now receives little attention.

2 Congruence and Genetics There are a number of Internet sites

that give instructions on making a DNA model by using string, toothpicks, and various colors of candy to represent the components of a DNA molecule.

3 Congruence and Literacy Johannes Gutenberg (ca. 1400–

1468) invented the fi rst printing press in the mid-1400s. Long before that time, however, the Chinese and Koreans had been printing both text and pictures using wood blocks and movable type made from porcelain and metal.

Advanced Student correctly provides all of the details asked for in the project as well as additional correct independent conclusions.

Profi cient Student correctly provides all of the details asked for in the project.

Partially profi cient Student correctly provides some of the details asked for in the project or provides all details with some inaccuracies.

Not profi cient Student correctly provides few of the details asked for in the project or provides all details with many inaccuracies.

No attempt Student makes little or no attempt to complete the project.

Project Rubric

SMP_TEGEO_C05PR_296-297.indd 296 4/11/08 10:50:07 AM

24 Call 1-800-648-2970

Chapter

Projects5

Chapter 5

1 Triangles on Curved SurfacesA special kind of non-Euclidean

geometry is called spherical geometry. In spherical geometry, the points are all on a sphere. To investigate this geometry you will need a sphere that you can write on, a washable marker, 3 or more pieces of string, and a protractor. a. Choose two points on the sphere. Stretch a

piece of string from one point to the other so that the string is as short as possible. This shortest path between two points can be thought of as a “line segment” in spherical geometry. A “line” through two points in spherical geometry is the path on the sphere that contains the shortest path from one point to another. If a line in spherical geometry is drawn, what is the result?

b. Create three points on the sphere. Create a “triangle” and measure its angles to the best of your ability. Do this for several different triangles. Make a conjecture about the sum of the measures of a triangle in spherical geometry.

2 Congruence and GeneticsHuman DNA is made up of four basic

building blocks. Any two blocks of the same kind should be congruent. Find out what these blocks are. Why do you think it is important that they be congruent? Make a model of a section of human DNA, illustrating the different building blocks.

3 Congruence and LiteracyJohannes Gutenberg is credited with

inventing the fi rst printing press—one of the most important inventions ever. The printing press produces many congruent copies of the same image. Find out how the printing press works, and prepare a presentation on this. Include some modern developments of this kind of machine, and how the methods of producing congruent images have changed over time.


296 Chapter 5

5Chapter

The projects relate to the content of the lessons of this chapter as follows:

Project Lesson(s)

1 5-6

2 5-1

3 5-1

4 5-4

5 5-5

6 5-7

1 Triangles on Curved Surfaces

Students may fi nd it interesting to check their ideas about angle measures on a sphere by consulting mathematics, geography, and cartography books. Spherical trigonometry was once a fairly common topic in textbooks, though it now receives little attention.

2 Congruence and Genetics There are a number of Internet sites

that give instructions on making a DNA model by using string, toothpicks, and various colors of candy to represent the components of a DNA molecule.

3 Congruence and Literacy Johannes Gutenberg (ca. 1400–

1468) invented the fi rst printing press in the mid-1400s. Long before that time, however, the Chinese and Koreans had been printing both text and pictures using wood blocks and movable type made from porcelain and metal.

Advanced Student correctly provides all of the details asked for in the project as well as additional correct independent conclusions.

Profi cient Student correctly provides all of the details asked for in the project.

Partially profi cient Student correctly provides some of the details asked for in the project or provides all details with some inaccuracies.

Not profi cient Student correctly provides few of the details asked for in the project or provides all details with many inaccuracies.

No attempt Student makes little or no attempt to complete the project.

Project Rubric


Projects

Projects 297

4 Proofs as Games Consider the following game: you are

given a certain number (say 15). At each step, you are allowed to add 2 to that number, or to divide it by 3 if it is divisible by 3 (so, if you started at 15, in the second step you could get to 17 or to 5). You are also given a target number (say 10). Your goal is to determine if you can start at the starting number and end at the target number.a. Is it possible to get from 15 to 9? How

many steps would you have to use? Explain your steps.

b. Is it possible to get from 15 to 2? Explain. c. Invent similar rules for a game, and try

to get from a starting number to a target number that you chose.

5 How Long Can Proofs Get? In this chapter, you encountered proofs

that were a few steps long. In mathematics, proofs can get to be extremely long. Use the Internet to fi nd out about a theorem with one of the longest proofs ever. How long was this proof? How long did it take to come up with? How many different people worked on it? Who were these people, and what were their different roles in the proof?

6 Star Polygons Drawn here are two star polygons. (Star

polygons are not polygons as we have defi ned “polygon.”)

A star polygon S can be formed from any convex n-gon provided n is odd. Draw diagonals from each vertex of P to the two vertices of P that are opposite it. You will wind up with n diagonals that form S. Each pair of consecutive diagonals form one of the n angles of S.a. Experiment with a DGS to fi nd the sum

of the measures of the angles of a star polygon of 5 sides.

b. Experiment with a DGS to fi nd the sum of the measures of the angles of a star polygon of 7 sides.

c. Make a conjecture from your experiments in Parts a and b and try to prove the conjecture.

Lester Wayne Mackey (left) and Brett Harrison (right) both researched the Seymour Conjecture while still students in high school. Mackey showed the graph theory conjecture was valid for some oriented graphs; Harrison later proved it valid for all of them.

Chapter Projects5

Projects 297

4 Proofs as Games You might want to have students

work in pairs or small groups on this project. When they have completed their games, have copies made to share with other class members.

5 How Long Can Proofs Get? Students can research books and

journals in the library or on the Internet to fi nd information on longest proofs. Have students share their fi ndings with the class.

6 Star Polygons Suggest that students experiment

with regular hexagons, regular octagons, and regular nonagons as well as the two star polygons given in the project.

Notes

Additional AnswersSample answers for Projects are in the Solution Manual in the Electronic Teacher’s Edition.




Chapter

Chapter 5

Summary and Vocabulary5

Vocabulary5-1*congruent fi gures* congruence

transformation*directly congruent*oppositely congruent

5-2corresponding parts

5-4interior anglesexterior anglesalternate interior anglesalternate exterior anglessame-side interior angles

5-6uniquely determinedauxiliary fi gurenon-Euclidean geometries*obtuse triangle*right triangle*acute triangle

5-7exterior angle of a polygon

Isometries preserve Angle measure, Betweenness, Collinearity, and Distance (A-B-C-D). As a result, any fi gure is the same size and shape as its image under an isometry. From this, we defi ne congruent figures as any two fi gures such that there is an isometry that maps one onto the other.

Congruence has some properties that are like those of equality: the refl exive, symmetric, and transitive properties. Three other basic properties of congruence are the Segment Congruence Theorem, the Angle Congruence Theorem, and the CPCF Theorem.

In a proof of a conditional statement p � q, p is the “given,” q is the “prove,” there is a drawing (when necessary), and a proof to show how q follows from p. Though mathematicians almost always write proofs in paragraphs, in elementary geometry, proofs are commonly written either in two columns or in paragraphs.

Most of the proofs in this chapter involve congruence. Common justifi cations in these proofs are defi nitions that involve segments of equal length or angles of equal measure, theorems about parallel lines, the congruence theorems, the Transitive Property of Congruence, and properties of refl ections.

The properties of refl ections help to prove that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. This explains why the construction of a circle through three noncollinear points works.

298 Chapter 5

Chapter

5

Summary and VocabularyThe Summary gives an overview of the entire chapter and provides an opportunity for students to consider the material as a whole. Thus, the Summary can be used to help students relate and unify the concepts presented in the chapter.

Vocabulary words and symbols are listed by lesson to provide a checklist of concepts that students must know. Emphasize to students that they should read the vocabulary list carefully before starting the Self-Test on page 300. If students do not understand the meaning of a vocabulary word, they should refer back to the indicated lesson.

Theorems and Properties covered in the chapter are listed below the Summary with page references included to lead students back to the location in the chapter where the theorem or property is stated.

298 Chapter 5

SMP_TEGEO_C05EOC_298-305.indd 298 4/11/08 10:44:26 AM

26 Call 1-800-648-2970

Chapter Wrap-Up

Summary and Vocabulary 299

From the Corresponding Angles Postulate we proved that two lines cut by a transversal are parallel if and only if a pair of alternate interior angles are congruent, a pair of alternate exterior angles are congruent, or a pair of same-side interior angles are supplementary. From this we also can deduce that there is exactly one line parallel to a given line through a point not on the line. This Uniqueness of Parallels Theorem helps deduce the Triangle-Sum Theorem, which is used to prove the Exterior Angle Theorem for Triangles, the Quadrilateral-Sum Theorem, and the formula S = (n - 2)180 for the sum, S, of the measures of the interior angles of any convex n-gon. This is then used to deduce the fact that the sum of the measures of one set of exterior angles is 360.

Postulates, Theorems, and Properties A-B-C-D Theorem (p. 252)Equivalence Properties of Congruence

(p. 254) Refl exive Property of Congruence Symmetric Property of Congruence Transitive Property of CongruenceSegment Congruence Theorem

(p. 258)Angle Congruence Theorem (p. 258)Corresponding Parts in Congruent

Figures (CPCF) Theorem (p. 259)Parallel Lines Theorem (p. 271)Alternate Interior Angles Theorem

(p. 272)Alternate Exterior Angles Theorem

(p. 272)

Same-Side Interior Angles Theorem (p. 272)

Perpendicular Bisector Theorem (p. 277)

Unique Circle Theorem (p. 283)Uniqueness of Parallels Theorem

(p. 284)Triangle-Sum Theorem (p. 285)Quadrilateral-Sum Theorem (p. 289)Polygon-Sum Theorem (p. 290)Exterior Angle Theorem for Triangles

(p. 290)Polygon Exterior Angle Theorem

(p. 291)

Chapter Summary & Vocabulary

5

Summary and Vocabulary 299



Chapter

Chapter 5

Summary and Vocabulary5

Vocabulary5-1*congruent fi gures* congruence

transformation*directly congruent*oppositely congruent

5-2corresponding parts

5-4interior anglesexterior anglesalternate interior anglesalternate exterior anglessame-side interior angles

5-6uniquely determinedauxiliary fi gurenon-Euclidean geometries*obtuse triangle*right triangle*acute triangle

5-7exterior angle of a polygon

Isometries preserve Angle measure, Betweenness, Collinearity, and Distance (A-B-C-D). As a result, any fi gure is the same size and shape as its image under an isometry. From this, we defi ne congruent figures as any two fi gures such that there is an isometry that maps one onto the other.

Congruence has some properties that are like those of equality: the refl exive, symmetric, and transitive properties. Three other basic properties of congruence are the Segment Congruence Theorem, the Angle Congruence Theorem, and the CPCF Theorem.

In a proof of a conditional statement p � q, p is the “given,” q is the “prove,” there is a drawing (when necessary), and a proof to show how q follows from p. Though mathematicians almost always write proofs in paragraphs, in elementary geometry, proofs are commonly written either in two columns or in paragraphs.

Most of the proofs in this chapter involve congruence. Common justifi cations in these proofs are defi nitions that involve segments of equal length or angles of equal measure, theorems about parallel lines, the congruence theorems, the Transitive Property of Congruence, and properties of refl ections.

The properties of refl ections help to prove that any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. This explains why the construction of a circle through three noncollinear points works.

298 Chapter 5

Chapter

5

Summary and VocabularyThe Summary gives an overview of the entire chapter and provides an opportunity for students to consider the material as a whole. Thus, the Summary can be used to help students relate and unify the concepts presented in the chapter.

Vocabulary words and symbols are listed by lesson to provide a checklist of concepts that students must know. Emphasize to students that they should read the vocabulary list carefully before starting the Self-Test on page 300. If students do not understand the meaning of a vocabulary word, they should refer back to the indicated lesson.

Theorems and Properties covered in the chapter are listed below the Summary with page references included to lead students back to the location in the chapter where the theorem or property is stated.

298 Chapter 5



WrightGroup.com



athematics Project

Teacher’s Edition


UnCh Sc MPr


V1

For more information, call 800-648-2970 or visit WrightGroup.com/UCSMP

1109 MA 09 W 7197

The University of Chicago School Mathematics Project:

• Upgrades student achievement in mathematics

• Updates the mathematics curriculum

• Increases the number of students who take

math beyond algebra and geometry

UCSMP engages students and maximizes student

learning through updated features such as:

• Problem-solving activities

• Connections to the real world

• Geometry, statistics, and probability in each course

• Activities and guided examples

• Daily review questions

• Technology that serves as a tool for discovery,

learning, and doing mathematics

Expect More. Achieve More.

Geometry Sampler

Documents

Transcript of Geometry Sampler