1. 1.If 4x = 20, then x = 5. 2.If tomorrow is Thursday, then today is Wednesday. Warm Up Please underline the hypothesis and circle the conclusion. Write the converse and if false provide a counterexample. 3.) Please prove the following: Given: 2x+4= 20 Prove: x=8

2. Connect Proofs to Section 2-4: Special Pairs of Angles 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.

3. 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 = 90 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.

4. 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 = 180 2. Definition of a straight angle 3. m  3 + m  4 = m  DEF 3. Angle Addition Postulate 2. m  DEF= 180 4. Substitution 1. m  DEF is a straight angle. 5. Definition of supplementary angles

5. 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

6. 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!! 1 3 2 4

7. Vertical Angle Theorem Proof Given:  1 and  2 are vertical angles. Prove:  1   2 1 3 2 4 ReasonsStatements 1.  1 and  2 are vertical  s. 1. Given 5.  1   2 2. m  1 + m  3 = 180 m  3 + m  2 = 180 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  1   24. Subtraction

8. Proof Example Given:  2   3; Prove:  1   4 1 32 4 ReasonsStatements 1.  2   3 1. Given 2.  2   1 3.  1   3 4.  3   4 5.  4   1 2. Vertical Angles are Congruent 4. Vertical Angles are Congruent 3. Substitution 5. Substitution You can also say “Vertical Angle Theorem” YOU CANNOT UNDER ANY CIRCUMSTANCES USE THE REASON “DEFINITION OF VERTICAL ANGLES” IN A PROOF!!

9. Proof ExampleGiven:  1 and  2 are supplementary;  3 and  4 are supplementary;  2   4 Prove:  1   3 1 3 2 4 1.  1 and  2 are supplementary  3 and  4 are supplementary 1. Given 2. m  1 + m  2 = 180 m  3 + m  4 = 180 2. Definition of Supplementary Angles 3. m  1 + m  2 = m  3 + m  43. Substitution 4.  2   4 or m  2 = m  44. Given 5. m  1 = m  3 or  1   35. Subtraction Property ReasonsStatements