Chapter 5: Relationships with Triangles

1. Chapter 5:Relationships with Triangles Section 5-4: Inverses, Contrapositves, and Indirect Reasoning

2. Objectives • To write the negation of a statement and find the inverse and contrapositve of a conditional statement. • To use indirect reasoning.

3. Vocabulary • Negation • Inverse • Contrapositive • Equivalent Statements • Indirect Reasoning • Indirect Proof

4. Negation • A negation of a statement has the opposite truth value.

5. Examples: • All triangles consist of 180º • True • Negation: All triangles do not consist of 180º • Bethlehem is the capital of Pennsylvania. • False • Negation: Bethlehem is not the capital of Pennsylvania. • We do not have school on Thanksgiving Day. • True • Negation: We have school on Thanksgiving Day.

6. Write the Negation • RABC is obtuse. • Lines m and n are not perpendicular.

7. Inverse • The inverse of the conditional “if p then q” is “if not p then not q” • The inverse negates both the hypothesis and conclusion.

8. Contrapositive • The contrapositive of the conditional “if p then q” is “if not q then not p” • The hypothesis of the conditional: • Switches the hypothesis and conclusion. • Negates both.

9. Write the Inverse and the Contrapositive • Conditional: • If a figure is a square, then it is a rectangle. • Inverse: • If a figure is not a square, then it is not a rectangle. • Contrapositive: • If a figure is not a rectangle, then it is not a square.

10. Recall: • A conditional and its converse can have different truth values. • Likewise, a conditional and its inverse can have different truth values. • The contrapositive will always have the same truth value as the conditional.

11. Equivalent Statements • Equivalent Statements have the same truth value. • Conditionals and contrapositives are equivalent.

12. Indirect Reasoning • In indirect reasoning, all possibilities are considered and then all but one is proved to be false. • The remaining possibility is true.

13. Example: • You are completing a geometry problem—finding the length of a triangle side. • You get the result: x2 = 16. • You think through the following steps: • You know that if x2 = 16, then x = 4 or x = -4. • You know the length of a side is not negative. • You conclude:___________

14. Indirect Proof • A proof involving indirect reasoning is an indirect proof. • In an indirect proof, there are often only two possibilities: • Statement • Negation

15. Writing an Indirect Proof: • Step One: State as an assumption, the opposite (negation) of what you want to prove: • Step Two: Show the assumption leads to a contradiction. • Step Three: Conclude that the assumption is false and what you want to prove must be true.

16. The first step of an indirect proof: • Prove: Quadrilateral QRWZ does not have four acute angles. • Assume: Quadrilateral QRWZ has four acute angles. • Prove: An integer n is divisible by 5. • Assume: An integer n is not divisible by 5.

17. Identifying Contradictions • Identify the two statements that contradict eachother: • VABC is acute • VABC is scalene • VABC is equiangular