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# Chapter 5.1 Write Indirect Proofs - PowerPoint PPT Presentation

Chapter 5.1 Write Indirect Proofs. Indirect Proofs are…?. An indirect Proof is used in a problem where a direct proof would be difficult to apply. It is used to contradict the given fact or a theorem or definition. D. Given: DB AC M is midpoint of AC Prove: AD ≠ CD. T. ~. A. C.

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Write Indirect Proofs

An indirect Proof is used in a problem where a direct proof would be difficult to apply.

It is used to contradict the given fact or a theorem or definition.

Given: DB ACM is midpoint of ACProve: AD ≠ CD

T

~

A

C

B

M

In order for AD and CD to be congruent, Δ ADC must be isosceles. But then the foot (point B) of the altitude from the vertex D and the midpoint M of the side opposite the vertex D would have to coincide. Therefore, AD ≠ DC unless point B point M.

• List the possibilities for the conclusion.

• Assume negation of the desired conclusion is correct.

• Write a chain of reasons until you reach an impossibility. This will be a contradiction of either:

• the given information or

• a theorem definition or known fact.

• State the remaining possibility as the desired conclusion.

Either RS bisects PRQ or RS does not bisect PRQ.

Assume RS bisects PRQ.

Then we can say that PRS  QRS.

Since RS PQ, we know that PSR  QSR.

Thus, ΔPSR  ΔQSR by ASA (SR  SR)

PR  QR by CPCTC.

But this is impossible because it contradicts the given fact that QR  PR. The assumption is false.

RS does not bisect PRQ.

T

Given:<H ≠ <KProve: JH ≠ JK

~

~

• Either JH is  to JK or it’s not.

• Assume JH is  to JK, then ΔHJK is isosceles because of congruent segments.

• Then  H is  to  K.

• Since  H isn’t congruent to  K, then JH isn’t congruentto JK.

J

H

K

Given: MATH is a squareIn terms of a, find M and A

What are the coordinates of A and M?

(2a, 2a)

A

M

(0,2a)

What is the area of MATH?

A = 4a2

What is the midpoint of MT?

T (2a, 0)

H (0, 0)

(a, a)