2 REVIEW: 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.)
3 EXAMPLE 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.
4 Given: 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.
5 Given: 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:
6 Vertical 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!!
7 Given: 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.
8 Given: 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
9 YOU CANNOT UNDER ANY CIRCUMSTANCES USE THE REASON “DEFINITION OF VERTICAL ANGLES” IN A PROOF!!
10 Ð1 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