Chapter 5.1 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. D 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.

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, 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

6. 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