2REVIEW: VOCABULARY from Section 2-4 Right Angle:An angle whose measure is 90.Straight Angle:An angle whose measure is 180.Complementary Angles:Two angles whose measures sum to 90.Supplementary Angles:Two angles whose measures sum to 180.Vertical Angles:The two non-adjacent angles that are created by a pair of intersecting lines. (They are across from one another.)
3EXAMPLE 1 Given: Ð1 and Ð2 are complementary Prove: ÐABC is a right angle.A12BCStatementsReasons1. Ð1 and Ð2 are complementary1. Given2. Definition of Complementary Angles2. mÐ1 + mÐ2 = 903. mÐ1 + mÐ2 = mÐABC3. Angle Addition Postulate4. mÐABC = 904. Substitution5. ÐABC is a right angle.5. Definition of a right angle.
4Given: ÐDEF is a straight angle. Prove: Ð3 and Ð4 are supplementary 3 EXAMPLE 2Given: ÐDEF is a straight angle.Prove: Ð3 and Ð4 are supplementary34DEFStatementsReasons1. mÐDEF is a straight angle.1. Given2. Definition of a straight angle2. mÐDEF= 1803. mÐ3 + mÐ4 = mÐDEF3. Angle Addition Postulate4. mÐ3 + mÐ4 = 1804. Substitution5. Definition of supplementary angles5. Ð3 and Ð4 are supplementary.
5Given: Prove: Vertical Angle Theorem: Vertical Angles are Congruent. Conditional: If two angles are vertical angles, then the angles are congruent.Given:Hypothesis: Two angles are vertical angles.Prove:Conclusion: The angles are congruent.Aside: Would the converse of this theorem work?If two angles are congruent,then the angles are vertical angles.FALSECounterexample:
6Vertical Angle Theorem Proof Prove: Ð2Given: Ð1 and Ð2 are vertical angles.1342NOTE: You cannot use the reason “Vertical Angle Theorem” or “Vertical Angles are Congruent” in this proof. That is what we are trying to prove!!
7Given: Ð1 and Ð2 are vertical angles. Vertical Angle Theorem ProofProve: Ð2Given: Ð1 and Ð2 are vertical angles.1342StatementsReasons1. Ð1 and Ð2 are vertical Ðs.1. Given2. mÐ1 + mÐ3 = 180mÐ3 + mÐ2 = 1802. Angle Addition Postulate3. mÐ1 + mÐ3 = mÐ3 + mÐ23. Substitution**. mÐ3 = mÐ3**. Reflexive Property4. mÐ1 = mÐ2 and Ð2 4. Subtraction4. mÐ1 = mÐ24. Subtraction Property5. Ð25. Definition Angles.
8Given: Ð2 @ Ð3; Prove: Ð1 @ Ð4 1. Ð2 @ Ð3 1. Given 2. Ð2 @ Ð1 EXAMPLE 31324Given: Ð3; Prove: Ð4StatementsReasons1. Ð31. GivenYou can also say“Vertical Angle Theorem”2. Ð12. Vertical Angles are Congruent3. Ð33. SubstitutionYou can also say“Vertical Angle Theorem”4. Ð44. Vertical Angles are Congruent5. Ð15. Substitution
9YOU CANNOT UNDER ANY CIRCUMSTANCES USE THE REASON “DEFINITION OF VERTICAL ANGLES” IN A PROOF!!
10Ð1 and Ð2 are supplementary; Ð3 and Ð4 are supplementary; Ð2 @ Ð4 Given:Ð1 and Ð2 are supplementary;Ð3 and Ð4 are supplementary;Ð4Prove: Ð31243StatementsReasons1. Ð1 and Ð2 are supplementaryÐ3 and Ð4 are supplementary1. Given2. mÐ1 + mÐ2 = 180mÐ3 + mÐ4 = 1802. Definition of Supplementary Angles3. mÐ1 + mÐ2 = mÐ3 + mÐ43. Substitution4. Ð4 or mÐ2 = mÐ44. Given5. mÐ1 = mÐ3 or Ð35. Subtraction Property