1. Chapter 6: Inequalities in Geometry 6.2 – Inverses and Contrapositives 6.3 – Indirect Proof (proof by contradiction) 6.4 – Triangle Inequalities

2. 6.2 – Inverses and Contrapositives 12/10 Conditional: If p, then q. Converse: If q, then p. Inverse: If not p, then not q. Contrapositive: If not q, then not p. Venn Diagram representation: Conditional: Contrapositive: A conditional statement and its contrapositive are logically equivalent. (True-True or False-False) qp TRUE False

3. 6.2 – Inverses and Contrapositives Conditional: If p, then q. Converse: If q, then p. Inverse: If not p, then not q. Contrapositive: If not q, then not p. Venn Diagram representation: Converse: Inverse: A conditional statement is not logically equivalent to its converse or its inverse. But, the converse and the inverse are logically equivalent statements. (True-True or False-False) pq TRUE False

4. 6.2 – Inverses and Contrapositives Example 1: Conditional: Converse: Inverse: Contrapositive: If you live in Castro Valley, then you live in CA. If you live in CA, then you live in Castro Valley. If you don’t live in Castro Valley, then you don’t live in CA. If you don’t live in CA, then you don’t live in Castro Valley. CACastro Valley Oakland San Francisco Los Angeles Hayward TRUE False TRUE

5. 6.2 – Inverses and Contrapositives Example 2: Conditional: Converse: Inverse: Contrapositive: If you are a runner, then you are an athlete. If you are an athlete, then you are a runner. If you are not a runner, then you are not an athlete. If you are not an athlete, then you are not a runner. Athlete Runner Cyclist Basketball player Football player Skier TRUE False TRUE

6. 6.2 – Inverses and Contrapositives Example 3: Conditional: Converse: Inverse: Contrapositive: If x 2 = 16. then x = 4. If x = 4, then x 2 = 16. If x 2 ≠ 16, then x ≠ 4. If x ≠ 4, then x 2 ≠ 16. x 2 = 16x = 4 x = -4 TRICKY: False TRUE False

7. 6.2 – Inverses and Contrapositives Example 4: Conditional: Converse: Inverse: Contrapositive: If a quadrilateral is a square, then it is regular. If a quadrilateral is regular, then it is a square. If a quadrilateral is not a square, then it is not regular. If a quadrilateral is not regular, then it is not a square. Square Regular Not Regular Not Square TRUE TRICKY: TRUE

8. 6.2 – Inverses and Contrapositives Example 5: (Page 211 W.E. #12) Given: A) Bridget Sullivan is a math teacher. B) August Campos assigns hours of homework. C) Andrew Byrnes assigns no homework at all. D) Jason Babler is not a math teacher. Bridget Sullivan assigns hours of HW. Math teachers assign hours of homework. (True Conditional) Andrew Byrnes is not a math teacher. NEI ( Not Enough Information ) Assign hours of HW Math Teacher No conclusion If you are a math teacher, then you assign hours of HW.

9. Classroom Practice Scavenger Hunt!