2.6 Proving Statements about Angles

1. 2.6 Proving Statements about Angles Geometry

2. Standards/Objectives Students will learn and apply geometric concepts. Objectives: • Use angle congruence properties • Prove properties about special pairs of angles.

3. Properties of Angle Congruence • Angle congruence is reflexive, symmetric, and transitive. • Reflexive: For any angle A, A ≅ A. • Symmetric: If A ≅ B, then B ≅ A • Transitive: If A ≅ B and B ≅ C, then A ≅ C.

4. Transitive Property of Angle Congruence Given: A ≅ B, B ≅ C Prove: A ≅ C C B A

5. Statement: A ≅ B, B ≅ C mA = mB mB = mC mA = mC A ≅ C Reason: Given Def. Cong. Angles Def. Cong. Angles Transitive property Def. Cong. Angles Transitive Property of Angle Congruence

6. Ex. 2: Using the Transitive Property Given: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3 Prove: m1 =40 1 4 2 3

7. Statement: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3 1 ≅ 3 m1 = m3 m1 = 40 Reason: Given Trans. Prop of Cong. Def. Cong. Angles Substitution Ex. 2:

8. All right angles are congruent. Given: 1 and 2 are right angles Prove: 1 ≅ 2

9. Statement: 1 and 2 are right angles m1 = 90, m2 = 90 m1 = m2 1 ≅ 2 Reason: Given Def. Right angle Transitive property Def. Cong. Angles

10. Properties of Special Pairs of Angles • Theorem 2.4: Congruent Supplements. If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If m1 +m2 = 180 AND m2 +m3 = 180, then 1 ≅ 3.

11. Congruent Complements Theorem • Theorem 2.5: If two angles are complementary to the same angle (or congruent angles), then the two angles are congruent. If m4 +m5 = 90 AND m5 +m6 = 90, then 4 ≅ 6.

12. Given: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 Prove: 2 ≅ 3 3 4 1 2

13. Statement: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 m 1 + m 2 = 180; m 3 + m 4 = 180 m 1 + m 2 =m 3 + m 4 m 1 = m 4 m 1 + m 2 =m 3 + m 1 m 2 =m 3 2 ≅ 3 Reason: Given Def. Supplementary angles Transitive property of equality Def. Congruent Angles Substitution property Subtraction property Def. Congruent Angles Ex. 4:

14. Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 2 m 1 + m 2 = 180

15. Using Linear Pairs m8 = m5 and m5 = 125. Explain how to show m7 = 55 7 8 5 6

16. Solution: • Using the transitive property of equality m8 = 125. The diagram shows that m 7 + m 8 = 180. Substitute 125 for m 8 to show m 7 = 55.

17. Vertical Angles Theorem • Vertical angles are congruent. 2 3 1 4 1≅ 3; 2≅ 4

18. Given: 5 and 6 are a linear pair, 6 and 7 are a linear pair Prove: 5 7 5 7 6

19. Statement: 5 and 6 are a linear pair, 6 and 7 are a linear pair 5 and 6 are supplementary, 6 and 7 are supplementary 5 ≅ 7 Reason: Given Linear Pair postulat Congruent Supplements Theorem Ex. 6: Proving Theorem 2.6