1. Prove Angle Pair Relationships
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
Give a reason for each statement.
1. If m = 90º and m = 90º, then m = m
ANSWER
Transitive Prop. of Eq.
2. If AB BC , then ABC is a right angle. ┴
ANSWER
Def. of perpendicular
3. If FG RS, then FG = RS =
ANSWER
Def. of segment congruence

3. Example 1 Write a proof. GIVEN: AB BC , DC BC PROVE: B C STATEMENTS
REASONS 1.
AB BC , DC BC 1.
Given 2.
B and C are right angles. 2.
Definition of perpendicular
lines 3.
B C 3.
Right Angles Congruence
Theorem

4. Example 2
Prove that two angles supplementary to the same angle are congruent.
GIVEN:
1 and are supplements.
3 and are supplements.
PROVE: 3

5. Example 2 STATEMENTS REASONS 1. 3 and 2 are supplements.
Given 1. 2.
m m = 180°
m m = 180° 2.
Definition of supplementary angles 3.
m m =
m m 2
Transitive Property of Equality 3. 4.
m = m
Subtraction Property of Equality 4. 5. 3
Definition of congruent angles 5.

6. Guided Practice
1. How many steps do you save in the proof in Example 1 by using the Right Angles Congruence Theorem?
ANSWER
2 Steps
2. Draw a diagram and write GIVEN and PROVE statements for a proof of each case of the Congruent Complements Theorem.

7. Guided Practice
2. Draw a diagram and write GIVEN and PROVE statements for a proof of each case of the Congruent Complements Theorem.
ANSWER 2 3 1
GIVEN: 2 and 3 are complementary to 1.
PROVE: 1 2 3 4
GIVEN: , 3 is complementary to 1, and 4 is complementary to 2.
PROVE:

8. Prove vertical angles are congruent. GIVEN:
Example 3
Prove vertical angles are congruent.
GIVEN:
5 and are vertical angles.
PROVE:
∠ 5 ∠ 7
STATEMENTS
REASONS
5 and are vertical angles. 1. 1.
Given 2.
5 and are a linear pair.
6 and are a linear pair. 2.
Definition of linear pair, as shown in the diagram 3.
5 and are supplementary.
6 and are supplementary. 3.
Linear Pair Postulate 4.
∠ 5 ∠
Congruent Supplements Theorem 4.

9. Guided Practice
In Exercises 3–5, use the diagram.
3. If m = 112°, find m 2, m 3, and m 4.
ANSWER
m = 68°
m = 112°
m = 68°
4. If m = 67°, find m 1, m 3, and m
ANSWER
m = 113°
m = 113°
m = 67°

10. Guided Practice
In Exercises 3–5, use the diagram.
5. If m = 71°, find m 1, m 2, and m
ANSWER
m = 109°
m = 71°
m = 109°

11. Guided Practice
6. Which previously proven theorem is used in Example 3 as a reason?
Congruent Supplements Theorem
ANSWER

12. Example 4
SOLUTION
Because TPQ and QPR form a linear pair, the sum of their measures is 180.
The correct answer is B.
ANSWER

13. Example 5
Tell whether the proof is logically valid. If it is not, explain how to change the proof so that it is valid.
GIVEN: 1 is a right angle.
PROVE: 3 is a right angle.
STATEMENTS
REASONS
is a right angle.
is a right angle.
1. Given
2. Vertical Angles Congruence Theorem
3. Right Angles Congruence Theorem

14. Example 5
SOLUTION
The proof is not logically valid. The Right Angles Congruence Theorem does not let you conclude that 3 is a right angle. It just says that all right angles are congruent.
Here is a way to complete the proof.

15. Example 5
REASONS
STATEMENTS
is a right angle.
1. Given
2. Vertical Angles Congruence Theorem
3. m 1 = m 3
3. Definition of congruent angles
4. m 1 = 90º
4. Definition of right angle
5. m 3 = 90º
5. Transitive Property of Equality
is a right angle.
6. Definition of right angle

16. Guided Practice
In Exercises 7 and 8, use the diagram.
7. Solve for x.
x = 49
ANSWER
8. Find m TPS.
m TPS = 148°
ANSWER

17. Guided Practice
9. Write a valid proof for Example 5 using the Congruent Supplements Theorem.
ANSWER
is a rt (Given)
is supp to 2; 2 is supp to (Two s that form a linear pair are supp)
(Congruent Supp Thm)
4. m 1 = m (Def of congruent s)
5. m 1 = 90º (Def of rt )
6. m 3 = 90º (Transitive Prop of Equality)
is a rt (Def of rt )

18. Lesson Quiz 1.
Give the reason for each step
GIVEN : 1 5
PROVE : 1
is supplementary to 4
STATEMENTS REASONS 2.
m 1 = m 5 3.
4 and are a linear pair. 5 1. 1
4 and are supplementary . 4.
m m = 180 5.
m m = 180 6. 7.
1 is supplementary to
Given
Def. of
Def. of linear pair
Linear Pair Post .
Def. of supplementary
Substitution Prop. of Eq.
Def. of supplementary