5-4 Indirect Proof

What is indirect reasoning?

Who uses indirect reasoning?

You wrote paragraph, two-column, and flow proofs.

• Write indirect algebraic proofs.

• Write indirect geometric proofs.

Direct Reasoning

In direct reasoning, you assume that the hypothesis is true and show that the conclusion must also be true.

If it is 3pm on a school day, then academic classes at Marian High School are finished for the day.

Indirect Reasoning

Indirect reasoning shows that a statement is true by proving that it cannot be false.

Assume the opposite—contradict it.

Indirect ReasoningMark’s car won’t start. He knows that

there are three likely reasons for this.1. His battery is dead2. His starter doesn’t work.3. He is out of gas.When a car’s starter needs to be replaced, the car

is silent when you try to start it. If the battery is dead, the engine “turns over” slowly, if at all. When Mark tries to start the car, it sounds normal. What do you think is wrong with his car?

Out of gas!

Three Key Steps in Indirect Reasoning.

Assume that the statement you are trying to prove is false.

Show that this assumption leads to a contradiction of something you know is true.

Conclude that your assumption was incorrect, so that the statement you originally wanted to prove must be true.

What would you assume for indirect reasoning?

If it rains, then I will wash my car. It rains and I do not wash my car.

.then,20If XYSTRm

.and,20 XYSTRm

State the Assumption for Starting an Indirect Proof

Answer: is a perpendicular bisector.

A. State the assumption you would make to start an indirect proof for the statement is not a perpendicular bisector.

A.

B.

C.

D.

Write an indirect proof to show that if –2x + 11 < 7, then x > 2.

Given: –2x + 11 < 7

Prove: x > 2

Step 1 Indirect Proof:

The negation of x > 2 is x ≤ 2. So, assume that x < 2 or x = 2 is true.

Step 2 Make a table with several possibilities for x assuming x < 2 or x = 2.

When x < 2, –2x + 11 > 7 and when x = 2, –2x + 11 = 7.

Step 2 Make a table with several possibilities for x assuming x < 2 or x = 2.

Step 3 In both cases, the assumption leads to a contradiction of the given information that –2x + 11 < 7. Therefore, the assumption that x ≤ 2 must be false, so the original conclusion that x > 2 must be true.

Which is the correct order of steps for the following indirect proof?

Given: x + 5 > 18

Prove: x > 13

I. In both cases, the assumption leads to a contradiction. Therefore, the assumption x ≤ 13 is false, so the original conclusion that x > 13 is true.

II. Assume x ≤ 13.

III.When x < 13, x + 5 = 18 and when x < 13, x + 5 < 18.

A. I, II, III

B. I, III, II

C. II, III, I

D. III, II, I

A. Yes, he can show by indirect proof that assuming that every sweater costs $32 or more leads to a contradiction.

B. No, assuming every sweater costs $32 or more does not lead to a contradiction.

SHOPPING David bought four new sweaters for a little under $135. The tax was $7, but the sweater costs varied.Can David show that at least one of the sweaters cost less than $32?

Given: ΔJKL with side lengths

5, 7, and 8 as shown.Prove: mK < mL

Write an indirect proof.

Step 3 Since the assumption leads to a contradiction, the assumption must be false. Therefore, mK < mL.

Indirect Proof:

Step 1 Assume that

Step 2 By angle-side relationships, By substitution, . This inequality is a false statement.

Which statement shows that the assumption leads to a contradiction for this indirect proof?

Given: ΔABC with side lengths 8, 10, and 12 as shown.

Prove: mC > mA

A. Assume mC ≥ mA + mB. By angle-side relationships, AB > BC + AC. Substituting, 12 ≥ 10 + 8 or 12 ≥ 18. This is a false statement.

B. Assume mC ≤ mA. By angle-side relationships, AB ≤ BC. Substituting, 12 ≤ 8. This is a false statement.

Who uses Indirect Reasoning?

Auto mechanicsPhysicians diagnosing diseasesCSILawyers

Eliminating possibilities that contradict a know fact can lead to the actual cause of a problem.

What is indirect reasoning?

In direct reasoning, you assume that the hypothesis is true and show that the conclusion must also be true.

Who uses indirect reasoning?

Auto mechanics, doctors, police, lawyers…

5-4 Assignment

Page 358, 11-20