Section 2.5

2-6 Proving Statements about Angles

C ONGRUENCE OF A NGLES THEOREM THEOREM 2.2 Properties of Angle Congruence Angle congruence is r ef lex ive, sy mme tric, 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

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

Transitive Property of Angle Congruence GIVEN A B, B CB C PROVE A CA C StatementsReasons mA = mB Definition of congruent angles 5 A  C Definition of congruent angles A  B,Given B  C mB = mC Definition of congruent angles mA = mC Transitive property of equality

Using the Transitive Property This two-column proof uses the Transitive Property. StatementsReasons m1 = m3 Definition of congruent angles GIVEN m3 = 40°,12,23 PROVE m1 = 40° 1 m1 = 40° Substitution property of equality 13 Transitive property of Congruence Givenm3 = 40°,12, 2323

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

Proving Theorem 2.3 StatementsReasons m1 = 90°, m2 = 90° Definition of right angles m1 = m2 Transitive property of equality 1  2 Definition of congruent angles GIVEN 1 and2 are right angles PROVE and2 are right angles Given

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

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

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 1 2 GIVEN 1 and2 are supplements PROVE and4 are supplements and2 are supplementsGiven 3 and4 are supplements 1  4 m1 + m2 = 180° Definition of supplementary angles m3 + m4 = 180°

Proving Theorem 2.4 StatementsReasons 3 GIVEN 1 and2 are supplements PROVE and4 are supplements m1 + m2 = Substitution property of equality m3 + m1 m1 + m2 = Transitive property of equality m3 + m4 m1 = m4 Definition of congruent angles

Proving Theorem 2.4 StatementsReasons GIVEN 1 and2 are supplements PROVE and4 are supplements m2 = m3 Subtraction property of equality 23 Definition of congruent angles

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

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

2 column proof form Given