Chapter 6 Inequalities in Geometry l 6 2

Chapter 6: Inequalities in Geometry l 6. 2 – Inverses and Contrapositives l 6. 3 – Indirect Proof (proof by contradiction) l 6. 4 – Triangle Inequalities

12/10 6. 2 – Inverses and Contrapositives • Conditional: • Converse: • Inverse: • Contrapositive: If p, then q. TRUE False If q, then p. If not p, then not q. False If not q, then not p. TRUE Venn Diagram representation: Conditional: Contrapositive: p q A conditional statement and its contrapositive are logically equivalent. (True-True or False-False)

6. 2 – Inverses and Contrapositives • Conditional: • Converse: • Inverse: • Contrapositive: If p, then q. If q, then p. TRUE False If not p, then not q. TRUE False If not q, then not p. Venn Diagram representation: Converse: q Inverse: p 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)

6. 2 – Inverses and Contrapositives Example 1: • Conditional: If you live in Castro Valley, then you live in CA. TRUE If you live in CA, then you live in • Converse: Castro Valley. False If you don’t live in Castro Valley, • Inverse: then you don’t live in CA. False If you don’t live in CA, then you • Contrapositive: don’t live in Castro Valley. TRUE Oakland Castro Valley San Francisco Los Angeles Hayward CA

6. 2 – Inverses and Contrapositives Example 2: If you are a runner, then you are an • Conditional: TRUE athlete. If you are an athlete, then you are • Converse: a runner. False If you are not a runner, then you • Inverse: are not an athlete. False If you are not an athlete, then you • Contrapositive: are not a runner. TRUE Cyclist Runner Basketball player Football player Skier Athlete

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

6. 2 – Inverses and Contrapositives Example 4: If a quadrilateral is regular, then it is a square. TRUE If a quadrilateral is a square, then • Converse: it is regular. TRUE If a quadrilateral is not regular, • Inverse: then it is not a square. TRUE • Contrapositive: If a quadrilateral is not a square, then it is not regular. TRUE • Conditional: TRICKY: Regular Not Regular Square Not Square

6. 2 – Inverses and Contrapositives Example 5: (Page 211 W. E. #12) • Given: Math teachers assign hours of homework. Assign (True Conditional) Math If you are a math teacher, then Teacher you assign hours of HW. • A) Bridget Sullivan is a math teacher. hours of HW Bridget Sullivan assigns hours of HW. • B) August Campos assigns hours of homework. NEI (Not Enough Information) • C) Andrew Byrnes assigns no homework at all. Andrew Byrnes is not a math teacher. • D) Jason Babler is not a math teacher. No conclusion

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