Write Indirect Proofs

1

• Chapter 5.1
• Write Indirect Proofs

2
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.

3
Given DB ACM is midpoint of ACProve AD
? CD
D
T

C
A
M
B
In order for AD and CD to be congruent, ? 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.
4
Rules

• 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.

5

• 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, ?PSR ? ?QSR 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
6
GivenltH ? ltKProve JH ? JK


J

• Either JH is ? to JK or its not.
• Assume JH is to JK, then ?HJK is isosceles
  because of congruent segments.
• Then ? H is ? to ? K.
• Since ? H isnt congruent to ? K, then JH isnt
  congruent to JK.

H
K