Matthew Gronsky Anthony Mercuri And Patrick Haggerty

1. Section 5.1 Indirect Proofs Matthew Gronsky Anthony Mercuri And Patrick Haggerty

2. Introduction 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.

3. Procedure • 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)

4. Sample Problems GIVEN: ∆ABC, And D is on but not at the midpoint PROVE: ∆ABD ∆ACD

5. Either ∆ ABD ∆ACD or ∆ABD ∆ACD Assume ∆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

6. Practice Problem A Given: ∆ABD with base is a median <BAC <DAC B D C Prove: ∆ ABD is not isosceles

7. 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.

8. Z Practice Problem 2 Y Given: ∆ ZOG with base , is an altitude to , <Z <O Prove: does not bisect G O

9. 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.

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

11. Works Cited 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.