2-5 Proving Angles Congruent

[Pages:6]2-5

1. Plan

Objectives

1 To prove and apply theorems about angles

Examples

1 Using the Vertical Angles Theorem

2 Proving Theorem 2-2

Math Background

Inductive reasoning can lead to conjectures, but only deductive reasoning can establish truth conclusively. Proving a theorem by logically progressing from a given condition to a necessary conclusion is a fundamental mathematical activity.

More Math Background: p. 78D

Lesson Planning and Resources

See p. 78E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to: Identifying Angle Pairs Lesson 1-6: Examples 4, 5, 6 Extra Skills, Word Problems, Proof

Practice, Ch. 1

2-5

Proving Angles Congruent

What You'll Learn

? To prove and apply

theorems about angles

. . . And Why

To find the measures of angles formed by the legs of a director's chair, as in Exercise 11

Check Skills You'll Need

x2 Algebra Find the value of each variable.

1. 50

2. 90

130 x

z

GO for Help Lesson 1-6

3. 35 y 55

Fill in each blank. 4. Perpendicular lines are two lines that intersect to form 9. right angles 5. An angle is formed by two rays with the same endpoint.

The endpoint is called the 9 of the angle. vertex

New Vocabulary ? theorem ? paragraph proof

1 Theorems About Angles

Key Concepts

Hands-On Activity: Vertical Angles

? Draw two intersecting lines. Number the angles as shown.

? Fold &1 onto &2. ? Fold &3 onto &4.

1

3

4

2

? Make a conjecture about vertical angles. Vertical angles are congruent.

You can use deductive reasoning to show that a conjecture is true. The set of steps you take is called a proof. The statement that you prove true is a theorem. The Investigation above leads to a conjecture that becomes the following theorem.

Theorem 2-1

Vertical Angles Theorem

Vertical angles are congruent.

&1 > &2 and &3 > &4

3

1 2

4

In the proof of a theorem, a "Given" list shows you what you know from the hypothesis of the theorem. You prove the conclusion of the theorem. A diagram records the given information visually.

110 Chapter 2 Reasoning and Proof

110

Special Needs L1 Have students perform the lesson Activity using lines that are not close to being perpendicular. Then have them cut out the angles and match them to reinforce the fact that vertical angles are congruent.

learning style: tactile

Below Level L2 Have students use protractors to verify the Vertical Angles Theorem. Actually measuring the angles also will reinforce the underlying algebraic argument in the paragraph proof on p. 111.

learning style: tactile

Here is what the start of many proofs will look like.

what you know

what you must show

Given: Prove:

diagram that shows what you know

Proof

There are many forms of proofs. A paragraph proof is written as sentences in a paragraph. Here is a paragraph proof of Theorem 2-1.

Given: &1 and &2 are vertical angles.

Prove: &1 > &2

d what you know S d what you show

31 2

Paragraph Proof: By the Angle Addition Postulate, m&1 + m&3 = 180 and m&2 + m&3 = 180. By substitution, m&1 + m&3 = m&2 + m&3. Subtract m&3 from each side. You get m&1 = m&2, or &1 > & 2.

You can use the Vertical Angles Theorem to solve for variables and find the measures of angles.

1 EXAMPLE Using the Vertical Angles Theorem

E

D

1 A 2 A

B B

C C C

D D

E E

B 3 A

E

C

D

B 4 A

5 A

B

E D C

E D C

B

Test-Taking Tip

Be sure you shade the correct ovals in the grid. Grading software reads only the shaded ovals, not the answer across the top.

Gridded Response Find the value of x. (4x) (3x 35)

35

. ./ ./ .

000 1111 2222 3333 4444 5555 6666 7777 8888 9999

4x = 3x + 35 Vertical angles are congruent.

x = 35

Subtract 3x from each side.

Quick Check

1 a. Find the measures of the labeled pair of vertical angles in the diagram above.

b. Find the measures of the other pair of vertical angles. 40; 40

140; 140

c. Check to see that adjacent angles are supplementary. 140 + 40 = 180

Key Concepts

The Vertical Angles Theorem is actually a special case of the following theorem. A proof of this theorem is shown on the next page. You can write a proof of another form of this theorem in Exercise 27.

Theorem 2-2

Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

2. Teach

Guided Instruction

Hands-On Activity

Provide patty paper (which is also used to separate frozen hamburger patties) for students to fold for this activity.

Technology Tip

Students can explore the Vertical Angles Theorem with geometry software.

Tactile Learners

Have students copy the diagrams for the Vertical Angles and Congruent Supplements Theorems to reinforce each Given and the conclusions that follow.

Alternative Method

Work as a class to write the paragraph proof preceding Example 1 in two columns, justifying each step in the right column as in Lesson 2-4.

1 EXAMPLE Teaching Tip Ask: How is this solution method like writing a paragraph proof? How is it different? Sample: You work step-by-step, building on what you know; you can quickly see the list of steps.

PowerPoint

Additional Examples

1 Find the value of x. 52

(2x + 3) (4x ? 101)

Lesson 2-5 Proving Angles Congruent 111

Advanced Learners L4 Have students prove the statement: "If two angles are congruent, then their supplements are congruent." This is the converse of the Congruent Supplements Theorem.

learning style: verbal

English Language Learners ELL Help students recognize that a proof is a set of steps to show a conjecture is true. Proofs use given information, definitions, postulates, and theorems as reasons that justify a step.

learning style: verbal

111

PowerPoint

Additional Examples

2 Write a paragraph proof using the given, what you are to prove, and the diagram.

WX

YZ

Given: WX = YZ Prove: WY = XZ Proof: It is given that

WX YZ. By adding the middle segment, XY, to each smaller

segment gives WX XY YZ XY. By the Segment

Addition Postulate, WX XY WY and XY YZ XZ. Using substitution, you get WY XZ.

Resources

? Daily Notetaking Guide 2-5 L3

? Daily Notetaking Guide 2-5--

Adapted Instruction

L1

Closure

Point out two things that are incorrect in the diagram. Explain your reasoning.

60? 120? 130?

50?

Pairs of vertical angles do not have the same measure; the Vertical Angles Theorem says they are congruent. One pair of supplementary angles has a sum of 170, and another pair has a sum of 190; supplementary angles have a sum of 180.

Proof 2 EXAMPLE Proving Theorem 2-2

Study what you are given, what you are to prove, and the diagram. Write a paragraph proof.

Given: &1 and &2 are supplementary. &3 and &2 are supplementary.

Prove: &1 > &3

1 2

3

Proof: By the definition of supplementary angles, m&1 + m&2 = 180 and m&3 + m&2 = 180. By substitution, m&1 + m&2 = m&3 + m&2. Subtract m&2 from each side. You get m&1 = m&3, or &1 > &3.

Quick Check 2 In the proof above, which Property of Equality allows you to subtract m&2 from

each side of the equation? Subtraction Property of Equality

Key Concepts

Theorem 2-3 is like the Congruent Supplements Theorem. You can demonstrate its proof in Exercises 7 and 28.

Theorem 2-3

Congruent Complements Theorem

If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Theorem 2-4 All right angles are congruent.

Theorem 2-5 If two angles are congruent and supplementary, then each is a right angle.

You can complete proofs of Theorems 2-4 and 2-5 in Exercises 14 and 21, respectively.

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

GO

for Help

Example 1 (page 111)

Find the value of each variable.

1.

2.

3x (80 - x)

y 3x 75

3. (x + 90) 4x

112

20

x 5 25, y 5 105

30

Find the measures of the labeled angles in each exercise.

4. Exercise 1 60, 60

5. Exercise 2 75, 105

6. Exercise 3 120, 120

Chapter 2 Reasoning and Proof

112

Example 2 (page 112)

B Apply Your Skills

Exercise 11

GO nline

Homework Help

Visit: Web Code: aue-0205

7. Developing Proof Complete this proof of one form of Theorem 2-3 by filling in the blanks.

If two angles are complements of the same angle, then the two angles are congruent.

Given: &1 and &2 are complementary.

&3 and &2 are complementary.

Prove: &1 > &3

1

Proof: By the definition of complementary angles, m&1 + m&2 = a9.09 and m&3 + m&2 = b. 9. 90

2 3

Then m&1 + m&2 = m&3 + m&2 by c. 9. substitution

Subtract m&2 from each side. You get m&1 = d. 9, or &1 > &3. ml3

8. Writing How is a theorem different from a postulate? Answers may vary. Sample: A theorem is proven and a postulate is assumed to be true.

9. Open-Ended Give an example of vertical angles in your home. Answers may vary. Sample: scissors

10. Reasoning Explain why this statement is true: If m&1 + m&2 = 180 and m&3 + m&2 = 180, then &1 > &3. See margin.

11. Design The two back legs of the director's chair pictured at the left meet in a 728 angle. Find the measure of each angle formed by the two back legs. See margin

3. Practice

Assignment Guide

1 A B 1-28 C Challenge

29-32

Test Prep Mixed Review

33-39 40-47

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 4, 7, 8, 20, 21.

Exercise 7 When they finish the proof, have students compare their justifications to those of the Vertical Angles Theorem proof.

x2 Algebra Find the value of each variable and the measure of each labeled angle.

x 5 14, y 5 15; 50, 50, 130

12.

13.

(x 10)

(4x 35)

(3x 8) (5x 20)

15; 25, 25

(5x 4y)

14. Developing Proof Complete this proof of Theorem 2-4 by filling in the blanks.

All right angles are congruent.

Given: &X and &Y are right angles.

Prove: &X > &Y

X

right angle

Y

Proof: By the definition of a. 9, m&X = 90 and m&Y = 90.

By the Substitution Property, m&X = b. 9, or &X > &Y. mlY

Oak St.

15. Multiple Choice What is the measure of the

angle formed by Park St. and Oak St.? C

35?

45?

55?

90?

Park St.

35

Name two pairs of congruent angles in each figure. Justify your answers. 16?18.

See margin.

16.

B

17.

18. K

D

E

F

O

A

C

I G

H

J

PL

M

Elm St. Main St.

? Pearson Education, Inc. All rights reserved.

GPS

Guided Problem

L3

Enrichment

Reteaching

Adapted Practice

PraNcamte ice

Practice 2-5

Find the values of the variables.

1. (3x 40)

(2x 10)

3.

(4z 10) z

5. (7x 3)

(4x 1) 65

L4

L2

L1

Class

Date

L3

Proving Angles Congruent

2.

(6y 10) (6y 10)

4. 32

(9x 4) 6.

(4y) (6y)

Write true or false. 7. &1 and &2 are vertical angles.

8. &2 and &3 are supplementary angles.

9. m&1 = m&3

10. m&3 + m&4 = 180

11. m&1 + m&3 = 180

12. &4 and &2 are adjacent angles.

Write three conclusions that can be drawn from each figure.

13. PO

125

M

N

Q

14.

C B

D

A

O

E

2

1

3

4

15. A

B

W

C

D

19. Coordinate Geometry &DOE contains poin) ts D(2, 3), O(0, 0), and E(5, 1). Find the coordinates of a point F so that OF is a side of an angle that is adjacent and supplementary to &DOE. Answers may vary. Sample: (?5, ?1)

Lesson 2-5 Proving Angles Congruent 113

113

Exercise 8 When students are finished, ask: How is a theorem similar to a postulate? Sample: Both are true statements about geometric figures. Point out that students will use both postulates and theorems that are already proved to prove each new theorem.

Connection to Architecture Exercise 9 Ask: What angles are

used frequently in house and building design? Sample: right angle, straight angle Have students discuss where these angles appear.

Exercise 15 Have students identify what they are trying to find in the diagram. Then ask: What angle information are you given? one right angle and a 35? angle

Exercises 16?18 Have partners explain their reasoning to each other. Point out that being able to explain your reasoning carefully is like writing a good proof in geometry.

Exercises 27, 28 Do these exercises as a class activity. Students may refer back to the proofs in the lesson for ideas, if necessary.

Connection to Algebra Exercises 30?32 Students must

set up and solve a system of two equations.

10. If ml1 ? ml2 180, and ml2 ? ml3 180, then ml1 ? ml2 ml2 ? ml3 by subst. Subtr. ml2 from each side ml1 ml3 or l1 O l3.

11. The two acute ' have measure 72. The two obtuse ' have measure 108.

16. lDOB O lAOC and lDOA O lBOC since they are vert. '.

17. lEIG O lFIH since all rt. ' are O; lEIF O lHIG since they are compl. of the same l.

114

GPS 20. Coordinate Geometry &AOX contains points A(1, 3), O(0, 0), and X(4, 0). a. Find the coordinates of a point B so that &BOA and &AOX are adjacent

complementary angles. a?b. See margin. )

b. Find the coordinates of a point C so that OC is a side of a different angle that is adjacent and complementary to &AOX.

21. Developing Proof Complete this proof of Theorem 2-5 by filling in the blanks.

If two angles are congruent and supplementary, then each is a right angle.

Given: &W and &V are congruent and supplementary.

Prove: &W and &V are right angles.

W

V

Proof: &W and &V are congruent, so m&W = m& a. 9. V &W and &V are supplementary so m&W + m&V = b. 9. 180 Substituting m&W for m&V, you get m&W + m&W = 180, or 2m&W = 180. By the c. 9 Property of Equality, m&W = 90. Division Since &W > &V, m&V = 90, too. Then both angles are d. 9 angles. right

31

24

Exercise 22

22. Sports In the photograph, the wheels of the racing wheelchair are tilted so that &1 > &2. What theorem can you use to justify the statement &3 > &4? Supplements of O ' are O.

x2 Algebra Find the measure of each angle.

23. &A is twice as large as its complement, &B. mlA 60, mlB 30

24. &A is half as large as its complement, &B. mlA 30, mlB 60

25. &A is twice as large as its supplement, &B. mlA 120, mlB 60

26. &A is half as large as twice its supplement, &B. mlA 90, mlB 90

Proof 27. Write a proof for this form of Theorem 2-2. See margin.

If two angles are supplements of congruent angles, then the two angles are congruent.

Given: &1 and &2 are supplementary. &3 and &4 are supplementary. &2 > &4

Prove: &1 > &3

1

2

3

4

Proof 28. Write a proof for this form of Theorem 2-3. See margin.

If two angles are complements of congruent angles, then the two angles are congruent.

Given: &1 and &2 are complementary. &3 and &4 are complementary. &2 > &4

Prove: &1 > &3

1 2

3 4

114

C Challenge

29. Paper Folding After you've done the Activity on page 110, answer these

questions. a?b. It is the bisector of both angles.

a. How is the first fold line you make related to angles 3 and 4?

b. How is the second fold line you make related to angles 1 and 2?

c. How are the two fold lines related to each other? Give a convincing

argument to support your answer. Sample: perpendicular; bisectors of

two adjacent supplementary angles form two adjacent angles whose

Chapter 2

Reasoning

and

measures Proof

add

to

12(180),

or

90.

18. lKPJ O lMPJ since they are marked O; lKPL O lMPL since they are suppl. of O '.

19. Answers may vary. Sample: (?5, ?1)

20. a. Answers may vary. B can be any point on the positive y-axis. Sample: (0, 5).

b. Answers may vary. Sample: (3, ?1)

x2 Algebra Find the value of each variable and the measure of each labeled angle.

30.

31.

(y + x)

2x

(y - x)

(x + y + 5) y 2x

32. 2x

4y (x y 10)

Test Prep

x 30, y 90; 60, 120, 60

x 35, y 70; 70, 110, 70

x 50, y 20; 80, 100, 80

Gridded Response

Find the measure of each angle. 33. an angle with measure 8 less than the measure of its complement 41 34. one angle of a pair of complementary vertical angles 45 35. an angle with measure three times the measure of its supplement 135

Use the diagram at the right to find the measure of each of the following angles.

36. &1 20

37. &2 90

38. &3 70

39. &4 110

70

41 32

Mixed Review

Lesson 2-4

GO

for Help

Use the given property to complete each statement.

40. Subtraction Property of Equality If 3x + 7 = 19, then 3x = 9. 12

41. Reflexive Property of Congruence AB > 9 AB

42. Substitution Property If MN = 3 and MN + NP = 15, then 9. 3 1 NP 5 15

Lesson 2-3

Use deductive reasoning to draw a conclusion. If not possible, write not possible.

43. If two lines intersect, then they are coplanar. Lines m and n are coplanar. not possible

44. If two angles are vertical angles, then they are congruent. &1 and &2 are vertical angles. l1 and l2 are congruent.

Lesson 2-2

45. If y 25, then y + 7 32. y + 7 32 if and only if y 25.

46. If you live south of the equator, then you live in Australia.

Each conditional statement below is true. Write its converse. If the converse is also true, combine the statements as a biconditional. 45. If y + 7 = 32, then y = 25.

46. If you live in Australia, then you live south of the equator. 47. If n . 0, then n2 . 0. If n2 S 0, then n S 0.

lesson quiz, , Web Code: aua-0205

Lesson 2-5 Proving Angles Congruent 115

27. By the def. of suppl. ', ml1 ? ml2 180 and ml3 ? ml4 180. By the Subst. Prop. ml1 ? ml2 ml3 ? ml4. It is given that l2 O l4, so ml2 ml4. Then by

the Subtr. Prop. of , ml1 ml3, or l1 O l3. 28. By the def. of compl. ', ml1 ? ml2 90 and ml3 ? ml4 90. By the Subst. Prop. of ,

ml1 ? ml2 ml3 ? ml4. It is given that l2 O l4, so ml2 ml4. Then by the Subtr. Prop. of , ml1 ml3 or l1 O l3.

4. Assess & Reteach

PowerPoint

Lesson Quiz

Use the diagram and mlABS 3x ? 6 and mlRBC 5x ? 20 for Exercises 1-4.

R

A

S

B

C

1. Find x. 13

2. Find m&ABS. 45? 3. Find m&SBC. 135?. 4. Without using the Vertical

Angle Theorem, what theorem can you use to prove that &ABR &SBC? Congruent Supplements Theorem

Alternative Assessment

Write this statement on the board. If two angles are congruent and complementary, then each has a measure of 45.

Have partners draw and label a diagram that fits the theorem and write a paragraph proof of the theorem.

Alternatively, if students are not yet ready to write a paragraph proof, ask them to write a full explanation of why the theorem makes sense. Writing an explanation is similar to writing a proof, but the language may be less intimidating to students.

Test Prep

A sheet of blank grids is available in the Test-Taking Strategies with Transparencies booklet. Give this sheet to students for practice with filling in the grids.

Resources For additional practice with a variety of test item formats: ? Standardized Test Prep, p. 121 ? Test-Taking Strategies, p. 116 ? Test-Taking Strategies with

Transparencies

115

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