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.

Given: DB AC M is midpoint of AC Prove: AD ≠ CD ~ 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.

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.

Either RS bisects PQR or RS does not bisect PQR 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 QR  PR. The assumption is false. RS does not bisect PRQ. T

Given:<H ≠ <K Prove: JH ≠ JK ~ ~ J 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. H K