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Name   Class    Date   

Geometric ProofGoing DeeperEssential question: How can you organize the deductive reasoning of a geometric proof?

You will use the Angle Addition Postulate and the following definitions to prove an important theorem about angles.

Opposite rays are two rays that have a common endpoint and form a straight line. A linear pair of angles is a pair of adjacent angles whose noncommon sides are opposite rays.

In the figure, __

› JK and

__ › JL are opposite rays; ∠MJK and ∠MJL are a linear

pair of angles.

Recall that two angles are complementary if the sum of their measures is 90°. Two angles are supplementary if the sum of their measures is 180°. The following theorem ties together some of the preceding ideas.

2-6

Linear Pair Theorem

If two angles form a linear pair, then they are supplementary.

Given: ∠MJK and ∠MJL are a linear pair of angles.

Prove: ∠MJK and ∠MJL are supplementary.

A Develop a plan for the proof.

Since it is given that ∠MJK and ∠MJL are a linear pair of angles, __

› JL and

__ › JK are opposite

rays. They form a straight angle. Explain why m∠MJK + m∠MJL must equal 180°.

B Complete the proof by writing the missing reasons. Choose from the following reasons.

p r o o f1

Statements Reasons

1. ∠MJK and ∠MJL are a linear pair. 1.

2. __

› JL and

__ › JK are opposite rays. 2. Definition of linear pair

3. __

› JL and

__ › JK form a straight line. 3.

4. m∠LJK = 180° 4. Definition of straight angle

5. m∠MJK + m∠MJL = m∠LJK 5.

6. m∠MJK + m∠MJL = 180° 6.

7. ∠MJK and ∠MJL are supplementary. 7. Definition of supplementary angles

Angle Addition Postulate Definition of opposite rays

Substitution Property of Equality Given

G-CO.3.9

M

J KL

M

L J K

Chapter 2 75 Lesson 6

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REFLECT

1a. Is it possible to prove the theorem by measuring ∠MJK and ∠MJL in the figure and showing that the sum of the angle measures is 180°? Explain.

1b. The proof shows that if two angles form a linear pair, then they are supplementary. Is this statement true in the other direction? That is, if two angles are supplementary, must they be a linear pair? Why or why not?

Statements Reasons

1. ∠VXW and ∠ZXY are vertical angles. 1.

2. ∠VXW and ∠ZXY are formed by intersecting lines.

2. Definition of vertical angles

3. ∠VXW and ∠WXZ are a linear pair.∠WXZ and ∠ZXY are a linear pair.

3. Definition of linear pair

4. ∠VXW and ∠WXZ are supplementary. 4.

5. m∠VXW + m∠WXZ = 180° 5.

6. 6. Linear Pair Theorem

7. 7. Definition of supplementary angles

8. m∠VXW + m∠WXZ = m∠WXZ + m∠ZXY 8. Transitive Property of Equality

9. m∠VXW = m∠ZXY 9.

p r a c t i c e

1. You can use the Linear Pair Theorem to prove a result about vertical angles. Complete the proof by writing the missing statements or reasons.

Given: ∠VXW and ∠ZXY are vertical angles, as shown.

Prove: m∠VXW = m∠ZXY

Y

Z

WX

V

Chapter 2 76 Lesson 6

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Practice Geometric Proof

Write a justification for each step.

Given: AB = EF, B is the midpoint of AC , and E is the midpoint of DF .

1. B is the midpoint of AC , and E is the midpoint of DF . _________________________

2. ≅AB BC , and ≅DE EF . _________________________

3. AB = BC, and DE = EF. _________________________

4. AB + BC = AC, and DE + EF = DF. _________________________

5. 2AB = AC, and 2EF = DF. _________________________

6. AB = EF _________________________

7. 2AB = 2EF _________________________

8. AC = DF _________________________

9. ≅AC DF _________________________

Fill in the blanks to complete the two-column proof. 10. Given: ∠HKJ is a straight angle.

KI bisects ∠HKJ. Prove: ∠IKJ is a right angle. Proof:

Statements Reasons

1. a._______________________________ 1. Given

2. m∠HKJ = 180° 2. b. ______________________________

3. c._______________________________ 3. Given

4. ∠IKJ ≅ ∠IKH 4. Def. of ∠ bisector

5. m∠IKJ = m∠IKH 5. Def. of ≅ s∠

6. d._______________________________ 6. ∠ Add. Post.

7. 2m∠IKJ = 180° 7. e. Subst. (Steps _______)

8. m∠IKJ = 90° 8. Div. Prop. of =

9. ∠IKJ is a right angle. 9. f. _______________________________

13

LESSON

2-6

CS10_G_MEPS710006_C02PWBL06.indd 13 4/21/11 5:58:19 PM

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2-6

Additional Practice

Name   Class    Date   

Chapter 2 77 Lesson 6

Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Problem Solving Geometric Proof

1. Refer to the diagram of the stained-glass window and use the given plan to write a two-column proof.

Given: ∠1 and ∠3 are supplementary. ∠2 and ∠4 are supplementary. ∠3 ≅ ∠4

Prove: ∠1 ≅ ∠2 Plan: Use the definition of supplementary angles to write

the given information in terms of angle measures. Then use the Substitution Property of Equality and the Subtraction Property of Equality to conclude that ∠1 ≅ ∠2.

The position of a sprinter at the starting blocks is shown in the diagram. Which statement can be proved using the given information? Choose the best answer. 2. Given: ∠1 and ∠4 are right angles.

A ∠3 ≅ ∠5 C m∠1 + m∠4 = 90° B ∠1 ≅ ∠4 D m∠3 + m∠5 = 180°

3. Given: ∠2 and ∠3 are supplementary. ∠2 and ∠5 are supplementary. F ∠3 ≅ ∠5 H ∠3 and ∠5 are complementary. G ∠2 ≅ ∠5 J ∠1 and ∠2 are supplementary.

97

LESSON

2-6

CS10_G_MEPS710006_C02PSL06.indd 97 5/19/11 10:58:17 PM

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Problem Solving

Chapter 2 78 Lesson 6