CONGRUENCE OF ANGLES THEOREM THEOREM 2.2 Properties of Angle Congruence Angle congruence is r ef lex ive, sy mme tric, and transitive. Here are some examples. REFLEX IVE For any angle A, A  A SYMMETRIC If A  B, then B  A TRANSITIVE If A  B and B  C, then A  C

Transitive Property of Angle Congruence Prove the Transitive Property of Congruence for angles. SOLUTION 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 C B C A C B

m A = m B Definition of congruent angles Transitive Property of Angle Congruence GIVEN A B, B C PROVE A C Statements Reasons A  B, Given B  C 1 2 m A = m B Definition of congruent angles 3 m B = m C Definition of congruent angles 4 m A = m C Transitive property of equality 5 A  C Definition of congruent angles

1 3 Transitive property of Congruence Using the Transitive Property This two-column proof uses the Transitive Property. GIVEN m 3 = 40°, 1 2, 2 3 PROVE m 1 = 40° Statements Reasons Given m 3 = 40°, 1 2, 2 3 1 2 1 3 Transitive property of Congruence 3 m 1 = m 3 Definition of congruent angles 4 m 1 = 40° Substitution property of equality

1 and 2 are right angles 1 2 You can prove Theorem 2.3 as shown. 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 and 2 are right angles PROVE 1 2

1 and 2 are right angles Given Proving Theorem 2.3 GIVEN 1 and 2 are right angles PROVE 1 2 Statements Reasons 1 and 2 are right angles Given 1 2 m 1 = 90°, m 2 = 90° Definition of right angles 3 m 1 = m 2 Transitive property of equality 4 1  2 Definition of congruent angles

PROPERTIES OF SPECIAL PAIRS OF ANGLES 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

If m 1 + m 2 = 180° m 2 + m 3 = 180° and then 1  3 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 3 1 2 3 1 If m 1 + m 2 = 180° m 2 + m 3 = 180° and then 1  3

PROPERTIES OF SPECIAL PAIRS OF ANGLES 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. 5 6 4

If m 4 + m 5 = 90° m 5 + m 6 = 90° and then 4  6 PROPERTIES OF SPECIAL PAIRS OF ANGLES 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. 6 5 4 6 4 If m 4 + m 5 = 90° m 5 + m 6 = 90° and then 4  6

1 and 2 are supplements Given Proving Theorem 2.4 GIVEN 1 and 2 are supplements 3 and 4 are supplements 1 4 PROVE 2 3 Statements Reasons 1 1 and 2 are supplements Given 3 and 4 are supplements 1  4 2 m 1 + m 2 = 180° Definition of supplementary angles m 3 + m 4 = 180°

m 1 + m 2 = Transitive property of equality m 3 + m 4 Proving Theorem 2.4 GIVEN 1 and 2 are supplements 3 and 4 are supplements 1 4 PROVE 2 3 Statements Reasons 3 m 1 + m 2 = Transitive property of equality m 3 + m 4 4 m 1 = m 4 Definition of congruent angles 5 m 1 + m 2 = Substitution property of equality m 3 + m 1

m 2 = m 3 Subtraction property of equality Proving Theorem 2.4 GIVEN 1 and 2 are supplements 3 and 4 are supplements 1 4 PROVE 2 3 Statements Reasons 6 m 2 = m 3 Subtraction property of equality 2 3 Definition of congruent angles 7

m 1 + m 2 = 180° PROPERTIES OF SPECIAL PAIRS OF ANGLES POSTULATE POSTULATE 12 Linear Pair Postulate If two angles for m a linear pair, then they are supplementary. m 1 + m 2 = 180°

1 3, 2 4 THEOREM Vertical angles are congruent Proving Theorem 2.6 THEOREM 2.6 Vertical Angles Theorem Vertical angles are congruent 1 3, 2 4

5 and 6 are a linear pair, Given 6 and 7 are a linear pair Proving Theorem 2.6 GIVEN 5 and 6 are a linear pair, 6 and 7 are a linear pair PROVE 5 7 Statements Reasons 1 5 and 6 are a linear pair, Given 6 and 7 are a linear pair 2 5 and 6 are supplementary, Linear Pair Postulate 6 and 7 are supplementary 3 5 7 Congruent Supplements Theorem