Chapter 5.1Chapter 5.1

Write Indirect Write Indirect ProofsProofs

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.

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

D

CMB

A

T

~

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.

Rules:

1. List the possibilities for the conclusion.2. Assume negation of the desired

conclusion is correct.3. Write a chain of reasons until you reach

an impossibility. This will be a contradiction of either:1. the given information or2. a theorem definition or known fact.

4. State the remaining possibility as the desired conclusion.

Either RS bisects PQR or RS does not bisect PQR.

Assume RS bisects PQR.

Then we can say that PRS QRS.

Since RS PQ, we know that PRS QSR.

Thus, ΔPSR ΔQSR by ASA (SR SR)

PR QR by CPCTC.

But this is impossible because it contradicts the given fact that PG 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 congruent to JK.

~

~

J

H K

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

T (2a, 0)H (0, 0)

M A

What is the area of MATH?

What is the midpoint of MT?

(2a, 2a)(0,2a)

What are the coordinates of A and M?

A = 4a2

(a, a)