GeometryGeometry

6.3 Indirect Proof6.3 Indirect Proof

Indirect ProofsIndirect Proofs

Up to this point, we have been Up to this point, we have been proving a statement true by proving a statement true by direct proofs.direct proofs.

Sometimes direct proofs are Sometimes direct proofs are difficult and we can instead prove difficult and we can instead prove a statement indirectly, which is a statement indirectly, which is very common in everyday logical very common in everyday logical thinking.thinking.

An Indirect Proof An Indirect Proof ExampleExample You say that my dog, Rex, dug a hole in your You say that my dog, Rex, dug a hole in your

yard on July 15yard on July 15thth. I will prove that Rex . I will prove that Rex did not did not dig a hole in your yard.dig a hole in your yard.

Let’s Let’s temporarily assumetemporarily assume that Rex that Rex diddid dig a dig a hole in your yard on July 15hole in your yard on July 15thth. . ThenThen he would have been in your yard on July he would have been in your yard on July 1515thth. . But this contradictsBut this contradicts the fact that Rex was in the fact that Rex was in the kennel from July 14the kennel from July 14thth to July 17 to July 17thth. I have . I have bills that show this is true. bills that show this is true. Thus, our assumption is false, thereforeThus, our assumption is false, therefore Rex Rex did not dig a hole in your backyard.did not dig a hole in your backyard.

Things to think about…Things to think about…

The negation of = is ≠, and vice versa.The negation of = is ≠, and vice versa.

The negation of > is ≤, and vice versa.The negation of > is ≤, and vice versa.

The key to an indirect proof is to know The key to an indirect proof is to know how to start it and to reason through it how to start it and to reason through it logically. logically.

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Example: Write the first step of an indirect proof.

a) If AB = BC, then ABC is not scalene. a.

b) If then . b.

c) If then XY // CD c.

An Indirect Proof An Indirect Proof TemplateTemplate1) Assume temporarily1) Assume temporarily that the that the

conclusion is not true.conclusion is not true.2) Then2) Then…Reason logically until you …Reason logically until you

reach a contradiction of a known reach a contradiction of a known fact or a given.fact or a given.

3) But this contradicts the given(or 3) But this contradicts the given(or fact) thatfact) that…state the contradiction.…state the contradiction.

4) Therefore . .4) Therefore . . , the conclusion is , the conclusion is true.true.

.

1. Given: Prove: and are not both right angles

m X m Y X Y

C

BA

T

SR

2. Given: AC = RT AB = RS

Prove: BC ST

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2n 4n 4. If > 6n, then .

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b

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2

1

5. Given:

Prove:

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P. 215-216 CE 1-10, WE 1-10, 14P. 215-216 CE 1-10, WE 1-10, 14

Given: nGiven: n22 > 6n > 6nProve: n ≠ 4Prove: n ≠ 4

Assume temporarilyAssume temporarily that n = 4. that n = 4.

ThenThen…n…n22 = 4 = 42 2 = 16 and 6n = 6(4) = = 16 and 6n = 6(4) = 24. Since 16 < 24, then n24. Since 16 < 24, then n22 < 6n. < 6n.

But this contradicts… But this contradicts… the given the given thatthat nn22 > 6n. > 6n.

Thus our assumption is false, . .Thus our assumption is false, . . n n ≠ 4.≠ 4.

.

Given: ≠ Given: ≠ Prove: ≠Prove: ≠

Assume temporarilyAssume temporarily that = . that = .

ThenThen…a//b because Corr. Angles …a//b because Corr. Angles Congruent Imply // Lines. Since a//b, Congruent Imply // Lines. Since a//b, then = because // lines then = because // lines imply that alt. int. angles are imply that alt. int. angles are congruent.congruent.

But this contradicts… But this contradicts… the given that the given that

≠≠

Thus our assumption is false, . .Thus our assumption is false, . . ≠ ≠

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1m 2m

1m

3m

b

a

1

23

1m1m

3m

3m

1m

1m

2m

2m

Given: Scalene Triangle Given: Scalene Triangle RENRENProve: ≠ Prove: ≠ Assume temporarily Assume temporarily = . = .

ThenThen…EN=RE by the Converse to the …EN=RE by the Converse to the Iso. Triangle Theorem. Thus, REN Iso. Triangle Theorem. Thus, REN would be an isosceles triangle.would be an isosceles triangle.

But this contradicts… But this contradicts… the given thatthe given that REN is a scalene triangle. REN is a scalene triangle.

Thus our assumption is false, . .Thus our assumption is false, . .

≠≠

.

Rm

Rm

Rm

Nm

Nm

Nm

RN

E