1. Sect. 2.6 Proving Statements about angles. Goal 1 Congruence of Angles Goal 2 Properties of Special Pairs of Angles.

2. THEOREM THEOREM 2.2 Properties of Angle Congruence Angle congruence is reflexive, symmetric, and transitive. Here are some examples. TRANSITIVE IfA  BandB  C, then A  C SYMMETRIC If A  B, then B  A REFLEX IVE For any angle A, A  A Congruence of Angles

3. Transitive Property of Angle Congruence Prove the Transitive Property of Congruence for angles. S OLUTION To prove the Transitive Property of Congruence for angles, begin by drawing three congruent angles. Label the vertices as A, B, and C. GIVEN A B, PROVE A CA C A B C B CB C Congruence of Angles

4. Proving Theorem 2.3 THEOREM THEOREM 2.3 Right Angle Congruence Theorem All right angles are congruent. You can prove Theorem 2.3 as shown. GIVEN 1 and2 are right angles PROVE 1212 Congruence of Angles

5. Proving Theorem 2.3 StatementsReasons 1 2 3 4 m1 = 90°, m2 = 90° Definition of right angles m1 = m2 Substitution 1  2 Definition of congruent angles GIVEN 1 and2 are right angles PROVE 1212 1 and2 are right angles Given Congruence of Angles

6. P ROPERTIES OF S PECIAL P AIRS OF A NGLES THEOREMS THEOREM 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 1 2 3

7. P ROPERTIES OF S PECIAL P AIRS OF A NGLES THEOREMS THEOREM 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 1 2 3 3 If m1 + m2 = 180° m2 + m3 = 180° and 1 then 1  3

8. THEOREMS THEOREM 2.5 Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 4 5 6 Properties of Special Pairs of Angles

9. P ROPERTIES OF S PECIAL P AIRS OF A NGLES THEOREMS THEOREM 2.5 Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 4 If m4 + m5 = 90° m5 + m6 = 90° and then 4  6 5 6 6 4

10. Proving Theorem 2.4 StatementsReasons 2 GIVEN 1 and2 are supplements PROVE 2323 3 and4 are supplements 1414 1 1 and2 are supplementsGiven 3 and4 are supplements m1 + m2 = 180° Definition of supplementary angles m3 + m4 = 180° Properties of Special Pairs of Angles 3 m1 + m2 = Substitution m3 + m4 4 1414 Given m1 = m4 Definition of congruent angles 5 6 m1 + m 2 = Substitution m 3 + m 1

11. Proving Theorem 2.4 StatementsReasons GIVEN 1 and2 are supplements PROVE 2323 3 and4 are supplements 1414 7 m2 = m3 Subtraction Pr. of Eq. 8 23 Definition of congruent angles Properties of Special Pairs of Angles

12. Beware of being lured into using the transitive property instead of the congruent supplements theorem.  A   BGiven  B   CGiven  A   CTrans. for   s  A supp to  BGiven  B supp to  CGiven  A supp to  CTrans.  1 comp to  2Given  2 comp to  3Given  1 comp to  3Trans.

13. POSTULATE POSTULATE 12 Linear Pair Postulate If two angles for m a linear pair, then they are supplementary. m1 + m2 = 180° P ROPERTIES OF S PECIAL P AIRS OF A NGLES

14. Proving Theorem 2.6 THEOREM THEOREM 2.6 Vertical Angles Theorem Vertical angles are congruent 1 3,24 Properties of Special Pairs of Angles

15. Proving Theorem 2.6 PROVE 5757 GIVEN 5 and6 are a linear pair, 6 and7 are a linear pair 1 2 3 StatementsReasons 5 and6 are a linear pair, Given 6 and7 are a linear pair 5 and6 are supplementary, Linear Pair Postulate 6 and7 are supplementary 5 7 Congruent Supplements Theorem Properties of Special Pairs of Angles

16. Homework 2.6 10, 12-17, 19-27 odd, 38-40