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

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

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

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

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

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

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 If m1 + m2 = 180° m2 + m3 = 180° and 1 then 1  3

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 Properties of Special Pairs of Angles

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 

Proving Theorem 2.4 StatementsReasons 2 GIVEN 1 and2 are supplements PROVE and4 are supplements 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 + m Given m1 = m4 Definition of congruent angles 5 6 m1 + m 2 = Substitution m 3 + m 1

Proving Theorem 2.4 StatementsReasons GIVEN 1 and2 are supplements PROVE and4 are supplements m2 = m3 Subtraction Pr. of Eq Definition of congruent angles Properties of Special Pairs of Angles

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.

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

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

Proving Theorem 2.6 PROVE 5757 GIVEN 5 and6 are a linear pair, 6 and7 are a linear pair 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

Homework , 12-17, odd, 38-40