1. Section 5-4: Indirect Reasoning March 7, 2012

2. Warm-up Warm-up: Practice 5-3: p. 58, 1-13

3. Warm-up

6. Questions on Homework?

7. Section 5-4: Indirect Reasoning  Objectives: Today you will learn to use indirect reasoning to prove a statement.

8. Section 5-4: Negation Negation – the negation of a statement has the opposite truth value.  Statement: “Benson is the capital of NC.”  Negation: “Benson is NOT the capital of NC.”  Statement: “ ∠ ABC is obtuse.”  Negation: “ ∠ ABC is not obtuse.”

9. Section 5-4: Inverse Inverse – negates both the hypothesis and the conclusion.  Statement: “If a figure is a square, then it is a rectangle.”  Inverse: (negate both): “If a figure is not a square, then it is not a rectangle.”  (Note: Inverses do not always have the same truth value as the original statement.)

10. Section 5-4: Contrapositive Contrapositive – switches the hypothesis and conclusion and negates them both.  Statement: “If a figure is a square, then it is a rectangle.”  Contrapositive: (switch and negate both): “If a figure is not a rectangle, then it is not a square.”  (Note: Contrapositives always have the same truth value as the original statement. They are equivalent statements.)

11. Section 5-4: Indirect Reasoning Indirect Reasoning: When all possibilities are considered and all but one are proved false. The remaining possibility must be true.

12. Section 5-4: Indirect Reasoning You get home from school and your brother tells you Hannah called. You know two Hannahs, but your brother doesn’t know which one called. You remember that one of the Hannahs you know has band practice after school, so it couldn’t have been her. It must have been the other Hannah.

13. Section 5-4: Indirect Proofs Indirect Proof: A proof that uses indirect reasoning. You know that a statement or its negation is true. Both cannot be true.

14. Section 5-4: Indirect Proofs Steps to Writing an Indirect Proof (p. 266) 1.Assume the opposite (negation) of what you want to prove. 2.Show that this assumption leads to a contradiction. 3.Conclude that the assumption must be false and that what you want to prove must be true.

15. Section 5-4: Indirect Proofs You get home from school and your brother tells you Hannah called. You know two Hannahs, but your brother doesn’t know which one called. Assumption: It was Hannah “A” Contradiction: You remember Hannah “A” has band practice after school, so it couldn’t have been her. Conclusion: It must have been the other Hannah.

16. First Step: Assume Opposite Example 1: Prove: “A triangle cannot have two right angles.” Assume: A triangle does have two right angles. Example 2: Prove: m ∠ A < m ∠ B Assume: m ∠ A ≥ m ∠ B

17. First Step: Assume Opposite Example 3: Prove: Quadrilateral QUIZ does not have four acute angles. Assume: Quadrilateral QUIZ does have four acute angles. Example 4: Prove: ΔABC ≅ ΔXYZ Assume: ΔABC ≆ ΔXYZ

18. Second Step: Identify Contradictions Example 5: I. ΔABC is an acute triangle. II. ΔABC is a scalene triangle. III. ΔABC is an equilateral triangle. II and III contradict each other

19. Second Step: Identify Contradictions Example 6: I. P, Q, and R are coplanar II. P, Q, and R are collinear III. m ∠ PQR = 60 II and III contradict each other

20. Second Step: Identify Contradictions Example 7: I. ΔABC is scalene II. m ∠ A < m ∠ B III. m ∠ A = m ∠ C I and III contradict each other

21. Second Step: Identify Contradictions Example 8: I. m ∠ A - m ∠ B = 0 II. m ∠ B < m ∠ A III. ∠ A and ∠ B are supplementary I and II contradict each other

22. Write an Indirect Proof: Example 9 Prove: ΔABC cannot contain 2 obtuse angles. 1. Assume the opposite of what you want to prove: ΔABC does contain 2 obtuse angles. Let ∠A and ∠B be obtuse. 2. Find contradiction: If ∠A and ∠B are obtuse, m∠A > 90 and m∠B > 90. So, m∠A + m∠B > 180. Since m∠C > 0, then m∠A + m∠B + m∠C > 180. This contradicts the Triangle Angle-Sum Theorem. 3. Conclusion: The assumption in Step 1 must be wrong. So, ΔABC cannot contain 2 obtuse angles

23. Write an Indirect Proof: Example 10

24. Write an Indirect Proof: Example 11

25. Wrap-up  Today you learned to use indirect reasoning to prove a statement.  Tomorrow you’ll learn about triangle inequalities.  Quiz on 5-1 to 5-3 tomorrow!  Homework p. 267-269, # 10-21, 31, 37, 38, 40 p. 272, 3rd column only: 3-33