1. Warm-Up Exercises Lesson 2.7, For use with pages 124-131 Give a reason for each statement. 1. If m 1 = 90º and m 2 = 90º, then m 1 = m 2. 2. If AB BC, then ABC is a right angle. ┴ 3. If FG RS, then FG = RS =

2. Objective: TSWBAT write two-column proofs involving angles. Homework: - Pg 127 #1-33 eoo

3. EXAMPLE 1 Use right angle congruence GIVEN: AB  BC, DC BC PROVE: B C Write a proof. STATEMENT REASONS 1. Given 2. Definition of perpendicular lines 3. Right Angles Congruence Theorem 2. B and C are right angles. 3. B C 1. AB  BC, DC BC

4. EXAMPLE 2 Prove a case of Congruent Supplements Theorem GIVEN: 1 and 2 are supplements. 3 and 2 are supplements. PROVE: 1 3 Prove that two angles supplementary to the same angle are congruent.

5. EXAMPLE 2 Prove a case of Congruent Supplements Theorem STATEMENT REASONS 1. 3 and 2 are supplements. 1 and 2 are supplements. Given 1. 2. m 1+ m 2 = 180° m 3+ m 2 = 180° 2. Definition of supplementary angles Substitution Property of Equality 3. m 1 + m 2 = m 3 + m 2 4. m 1 = m 3 5.1 3 Subtraction Property of Equality 4. Definition of congruent angles 5.

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

7. GUIDED PRACTICE for Examples 1 and 2 Write a proof. Given: 1 and 3 are complements; 3 and 5 are complements. Prove:  1 5 ANSWER

8. GUIDED PRACTICE for Examples 1 and 2 Statements (Reasons) 1. 1 and 3 are complements; 3 and 5 are complements. (Given) 2.  1 5 Congruent Complements Theorem.

9. EXAMPLE 3 Prove the Vertical Angles Congruence Theorem GIVEN: 5 and 7 are vertical angles. PROVE:  5  7 Prove vertical angles are congruent.

10. EXAMPLE 3 Prove the Vertical Angles Congruence Theorem 5 and 7 are vertical angles. 1. STATEMENT REASONS 1. Given 2. 5 and 6 are a linear pair. 6 and 7 are a linear pair. 2. Definition of linear pair, as shown in the diagram 3. 5 and 6 are supplementary. 6 and 7 are supplementary. 3. Linear Pair Postulate 4.  5  7 Congruent Supplements Theorem 4.

11. GUIDED PRACTICE for Example 3 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°

12. GUIDED PRACTICE for Example 3 4. If m 2 = 67°, find m 1, m 3, and m 4. ANSWER m 1 = 113° m 3 = 113° m 4 = 67° 5. If m 4 = 71°, find m 1, m 2, and m 3. ANSWER m 1 = 109° m 2 = 71° m 3 = 109°

13. GUIDED PRACTICE for Example 3 6. Which previously proven theorem is used in Example 3 as a reason? Congruent Supplements Theorem ANSWER

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

15. GUIDED PRACTICE for Example 4 7. Solve for x. SOLUTION Because TPQ and QPR form a linear pair, the sum of their measures is 180°. The correct answer is B. 32 + (3x +1) = 180 Original equation 32 + 3x +1 = 180 Distributive property of equality 3x = 147 Subtract 33 from each side x = 49 Divide each side by 3 Use the diagram in Example 4.

16. GUIDED PRACTICE for Example 4 8. Find m TPS. m TPS = (3x + 1)° Substitute the value x = 49 m TPS = (147 +1)° m TPS = 148° SOLUTION Use the diagram in Example 4. m TPS = (3 49 +1)°

17. Closure Properties of Congruence