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C ONGRUENCE OF A NGLES

C ONGRUENCE OF A NGLES. For any angle A, A  A. REFLEX IVE. If A  B, then B  A. SYMMETRIC. If A  B and B  C, then A  C. TRANSITIVE. THEOREM. THEOREM 2.2 Properties of Angle Congruence.

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C ONGRUENCE OF A NGLES

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  1. CONGRUENCE OF ANGLES For any angle A, A  A REFLEXIVE If A B, then BA SYMMETRIC If A  B and B  C, then A  C TRANSITIVE THEOREM THEOREM 2.2Properties of Angle Congruence Angle congruence is reflexive, symmetric, and transitive. Here are some examples.

  2. Transitive Property of Angle Congruence C B AB, AC GIVEN PROVE A BC 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.

  3. Transitive Property of Angle Congruence 5 4 3 1 2 Statements Reasons A  B, Given AB, AC GIVEN PROVE B  C BC mA = m B Definition of congruent angles mB = m C Definition of congruent angles mA = m C Transitive property of equality A CDefinition of congruent angles

  4. Using the Transitive Property m 3 = 40°, 12, 23 GIVEN m 1 = 40° PROVE 4 1 2 3 Statements Reasons m 3 = 40°, 1 2, Given 2 3 1 3Transitive property of Congruence m 1 = m 3 Definition of congruent angles m 1 = 40° Substitution property of equality This two-column proof uses the Transitive Property.

  5. Proving Theorem 2.3 1 and 2 are right angles GIVEN 1 2 PROVE THEOREM THEOREM 2.3Right Angle Congruence Theorem All right angles are congruent. You can prove Theorem 2.3 as shown.

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

  7. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.4Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. 2 1 3

  8. 1 2 3 1 and 3 If m 1 + m 2 = 180° m 2 + m 3 = 180° 1  3 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.4Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles) then they are congruent. then

  9. PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.5Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. 5 6 4

  10. 5 6 4 6 4 and If m 4 + m 5 = 90° m 5 + m 6 = 90° 4  6 PROPERTIES OF SPECIAL PAIRS OF ANGLES THEOREMS THEOREM 2.5Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. then

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

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

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

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

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

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

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