Matthew Gronsky

Anthony Mercuri

And

Patrick Haggerty

Section 5.1 Indirect Proofs

Indirect proofs are proofs that are not in the traditional format of two-column proofs.

Indirect proofs are useful when you need to prove a negative. For example;

does not bisect < ABD.

Introduction

AB

List the possibilities for the conclusion. Assume the opposite of the desired conclusion is correct. Write a chain of reasons until you reach an impossibility.

This will be a contradiction of either Given information A theorem, definition, or other known fact.

State the remaining possibility as the desired conclusion.

(Taken from Geometry for Enjoyment and Challenge)

Procedure

GIVEN: ∆ABC, And D is on but not at the midpoint

PROVE: ∆ABD ∆ACD

Sample Problems

ABACBC

Either ∆ ABD ∆ACD or ∆ABD ∆ACDAssume ∆ABD ∆ACD therefore,(CPCTC)But this contradicts the given fact that D is on

but not at the midpoint .Therefore the assumption is false and it follows that ∆ABD ∆ACD because that is the only other

possibility

BD CD

BC

A

BCD

Given: ∆ABD with base DB

Prove: ∆ ABD is not isosceles

AC is a median

<BAC <DAC

Practice Problem

Either ∆ABD is isosceles or ∆ABD is not isosceles Assume ∆ABD is not isosceles. So . <D <B. It is given that ∆ABD with base . Also, is a median.So (If a line that extends from a ∆ vertex is a

median then it divides the opp side into 2 segs.Hence, ∆ACD ∆ACB (SAS). <BAC <DAC (CPCTC).

But this is impossible b/c it contradicts the given fact that <BAC <DAC.

Therefore, the assumption is false and it follows that ∆ABD is not isosceles b/c that is the only other possibility.

AB AD DB AC

DC CB

Given: ∆ ZOG with base , is an altitude to , <Z <O

Prove: does not bisect

Practice Problem 2Z

OG

Y

GY

GO

ZO

GY

ZO

Either bisects or does not bisect Assume bisects . Therefore (If a ray

bisects a segment, it divides the segment into 2 congruent segments.)

It is given that ∆ZOG with base and is an altitude to . So <ZYG and < OYG are right angles (If a line extending from a triangle vertex is an altitude then it forms right angles with the opposite side). <ZYG <OYG (Right <s are congruent). (Reflexive Property). Therefore ∆YOG ∆YZG (SAS). Hence, <Z <O by CPCTC.

But this is impossible b/c it contradicts the given fact that <Z <O.

ZOGYZO

GYZO

OYZY

GY

GY

GY

ZO

GY

GO

the assumption is false and it follows that does not bisect b/c that is the only other possibility.

GYZO

Indirect Proofs. Golden Plains School Division, 2009. Web. January 17, 2010.

Rhoad, Richard. Geometry for Enjoyment and Challenge: New Edition. New York: McDougal Little & Company, 1991.

Works Cited