Prove Angle Pair Relationships Warm Up Lesson Presentation Lesson Quiz

Warm-Up Give a reason for each statement. 1. If m 1 = 90º and m 2 = 90º, then m 1 = m 2. 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

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

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

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

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.

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: 2 3 1 2 3 4 GIVEN: 1 2, 3 is complementary to 1, and 4 is complementary to 2. PROVE: 3 4

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

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

Guided Practice In Exercises 3–5, use the diagram. 5. If m 4 = 71°, find m 1, m 2, and m 3. ANSWER m 1 = 109° m 2 = 71° m 3 = 109°

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

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

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 3. 3 is a right angle. 1. 1 is a right angle. 2. 1 3 1. Given 2. Vertical Angles Congruence Theorem 3. Right Angles Congruence Theorem

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.

Example 5 REASONS STATEMENTS 1. 1 is a right angle. 1. Given 2. 1 3 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 6. 3 is a right angle. 6. Definition of right angle

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

Guided Practice 9. Write a valid proof for Example 5 using the Congruent Supplements Theorem. ANSWER 1. 1 is a rt (Given) 2. 1 is supp to 2; 2 is supp to 3 (Two s that form a linear pair are supp) 3. 1 3 (Congruent Supp Thm) 4. m 1 = m 3 (Def of congruent s) 5. m 1 = 90º (Def of rt ) 6. m 3 = 90º (Transitive Prop of Equality) 7. 3 is a rt (Def of rt )

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 4 + m 5 = 180 5. m 4 + m 1 = 180 6. 7. 1 is supplementary to 4. Given Def. of Def. of linear pair Linear Pair Post . Def. of supplementary Substitution Prop. of Eq. Def. of supplementary