Indirect Proofs. State the hypothesis and the conclusion of the following conditional statement:

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1 State the hypothesis and the conclusion of the following conditional statement: If it is cold outside, then Mr. Bates will wear a coat. Write your own conditional statement and state the hypothesis and the conclusion. 1

2 What are the first steps when writing a proof (either a two column or a flowchart proof)? What is the last step when writing a proof? 2

3 Every conditional statement written in the form If p, then q has three additional conditional statements associated with it: converse, contrapositive, and inverse. Recall from previous lessons, to state the converse, reverse the hypothesis, p, and the conclusion, q. To state the inverse, negate both parts. To state the contrapositive, negate each part and reverse them. 3

4 Conditional Statement Converse Inverse Contrapositive 4

5 For each conditional statement written in propositional form, identify the hypothesis p and the conclusion q. Identify the negation of the hypothesis and conclusion, and then write the inverse and contrapositive of the conditional statement. 1.If a quadrilateral is a square, then the quadrilateral is a rectangle. a. Hypothesis p: b. Conclusion q: c. Is the conditional statement true? Explain. d. Not p: e. Not q: f. Inverse: g. Is the inverse true? Explain. h. Contrapositive: i. Is the contrapositive true? Explain. 5

6 If an integer is even, then the integer is divisible by two. a. Hypothesis p: b. Conclusion q: c. Is the conditional statement true? Explain. d. Not p: e. Not q: f. Inverse: g. Is the inverse true? Explain. h. Contrapositive: i. Is the contrapositive true? Explain. 6

7 If a polygon has six sides, then the polygon is a pentagon. a. Hypothesis p: b. Conclusion q: c. Is the conditional statement true? Explain. d. Not p: e. Not q: f. Inverse: g. Is the inverse true? Explain. h. Contrapositive: i. Is the contrapositive true? Explain. 7

8 What do you notice about the truth value of a conditional statement and the truth value of its inverse? What do you notice about the truth value of a conditional statement and the truth value of its contrapositive? 8

9 All of the proofs up to this point were direct proofs. A direct proof begins with the given information and works to the desired conclusion directly through the use of givens, definitions, properties, postulates, and theorems. An indirect proof is different and may be shorter than a direct proof. An indirect proof, or proof by contradiction, uses the contrapositive. If you prove the contrapositive true, then the statement is true. Begin by assuming the conclusion is false and use this assumption to show one of the given statements is false, thereby creating a contradiction. 9

10 In an indirect proof: State the assumption; use the negation of the conclusion or prove statement. Write the givens. Write the negation of the conclusion. Use the assumption, in conjunction with definitions, properties, postulates, and theorems, to prove a given statement is false, thus creating a contradiction. 10

11 Write the negation of the conclusion. Use the assumption, in conjunction with definitions, properties, postulates, and theorems, to prove a given statement is false, thus creating a contradiction. Hence, your assumption leads to a contradiction; therefore, the assumption must be false. This proves the contrapositive. Let s look at an example of an indirect proof. C Given: In!CHT, CH! CT CA does not bisect HT Prove:!CHA "!CTA H e Note, you are to prove "!CTA. ssume the on of this ment,!!cta. ecomes the A Statements 1.!CHA!!CTA Reasons 1. Assumption 2. CA does not bisect HT 2. Given 3. HA! TA 3. CPCTC T 4. CA bisects HT 4. Definition of bisect 5.!CHA!!CTA is false 5. This is a contradiction. Step 4 contradicts step 2; 11

12 R B Statements N Reasons 2. When writing an indirect proof, it is often easier to write it as a paragraph proof. Write the proof in Question 1 as a paragraph proof. 12

13 Now try one yourself! A 1. Given: BR bisects!abn!bra!!brn Prove: AB! NB R B Statements N Reasons 13

14 R B Statements N Reasons 2. When writing an indirect proof, it is often easier to write it as a paragraph proof. Write the proof in Question 1 as a paragraph proof. 14

15 28. Given: ET! DT and EU # DU T Prove: EX # DX E X U D 2010 Carnegie Learning, Inc. 6 15

16 R B Statements N Reasons 2. When writing an indirect proof, it is often easier to write it as a paragraph proof. Write the proof in Question 1 as a paragraph proof. 16