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REVIEW: VOCABULARY from Section 2-4 Complementary Angles: Supplementary Angles: Vertical Angles: Two angles whose measures sum to 90. Two angles whose measures sum to 180. The two non-adjacent angles that are created by a pair of intersecting lines. (They are across from one another.) Right Angle: An angle whose measure is 90. Straight Angle: An angle whose measure is 180.

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EXAMPLE 1 Given: 1 and 2 are complementary Prove: ABC is a right angle. A B C 1 2 StatementsReasons 1. 1 and 2 are complementary 1. Given 2. m 1 + m 2 = Definition of Complementary Angles 3. m 1 + m 2 = m ABC 3. Angle Addition Postulate 4. m ABC = 904. Substitution 5. ABC is a right angle.5. Definition of a right angle.

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Given: DEF is a straight angle. Prove: 3 and 4 are supplementary 34 DEF StatementsReasons 5. 3 and 4 are supplementary. 1. Given 4. m 3 + m 4 = Definition of a straight angle 3. m 3 + m 4 = m DEF 3. Angle Addition Postulate 2. m DEF= Substitution 1. m DEF is a straight angle. 5. Definition of supplementary angles EXAMPLE 2

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Vertical Angle Theorem: Vertical Angles are Congruent. Hypothesis: Two angles are vertical angles. Conclusion: The angles are congruent. Conditional: If two angles are vertical angles, then the angles are congruent. Given: Prove: Aside: Would the converse of this theorem work? If two angles are congruent, then the angles are vertical angles. Counterexample: FALSE

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Vertical Angle Theorem Proof Given: 1 and 2 are vertical angles. Prove: 1 2 NOTE: You cannot use the reason Vertical Angle Theorem or Vertical Angles are Congruent in this proof. That is what we are trying to prove!!

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Vertical Angle Theorem Proof Given: 1 and 2 are vertical angles. Prove: ReasonsStatements 1. 1 and 2 are vertical s. 1. Given m 1 + m 3 = 180 m 3 + m 2 = Angle Addition Postulate 3. m 1 + m 3 = m 3 + m 2 3. Substitution **. m 3 = m 3 **. Reflexive Property 4. m 1 = m 2 4. Subtraction Property 5. Definition of Angles. 4. m 1 = m 2 and Subtraction

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EXAMPLE 3 Given: 2 3; Prove: ReasonsStatements Given Vertical Angles are Congruent 4. Vertical Angles are Congruent 3. Substitution 5. Substitution You can also say Vertical Angle Theorem You can also say Vertical Angle Theorem

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YOU CANNOT UNDER ANY CIRCUMSTANCES USE THE REASON DEFINITION OF VERTICAL ANGLES IN A PROOF!!

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Given: 1 and 2 are supplementary; 3 and 4 are supplementary; 2 4 Prove: and 2 are supplementary 3 and 4 are supplementary 1. Given 2. m 1 + m 2 = 180 m 3 + m 4 = Definition of Supplementary Angles 3. m 1 + m 2 = m 3 + m 43. Substitution or m 2 = m 44. Given 5. m 1 = m 3 or Subtraction Property ReasonsStatements

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