Multiple converses to a theorem. Let us see how multiple statements can be considered converses to the Midpoint Theorem and how Theorem 7 is one of them. To formulate a converse we should express the original statement in “If … then …” form. For a complex statement, there may be multiple ways to do that. The Midpoint Theorem starts with △ABC and points P and Q on sides AB and AC respectively. The theorem has two assumptions and two conclusions, which we have named for further discussion. Assumptions: (P MID) P is the midpoint of AB and (Q MID) Q is the midpoint of AC. Conclusions: (PRLL) PQ ∥ BC and (HALF) PQ = BC/2. Let us use these four named conditions to discuss various possible statements. (i) A natural converse of the Midpoint Theorem would be: “If (PRLL) and (HALF) are true, then (P MID) and (Q MID) are true.” Write out this statement fully. Experiment and see that it seems to be true! Prove the statement. (Hint: Try to run a proof of the Midpoint Theorem backwards.) (ii) To see Theorem 7 as a converse of the Midpoint Theorem, we first write the Midpoint Theorem as follows: “Suppose P is the midpoint of side AB of △ABC and Q is a point on side AC. If Q is the midpoint of AC then PQ ∥ BC.” (Also PQ = BC/2, but let us set that aside for now.) Write the converse of the second sentence in quotes by keeping the first sentence the same. Verify that Theorem 7 is what you get! (And PQ = BC/2 also follows.) Now write Theorem 7 in terms of the four named conditions. (iii) The discussion so far suggests that it is reasonable to take two of the four listed statements as assumptions and ask if the other two are true. Verify that only one such combination remains to be examined. “If (P MID) and (HALF) are true, then can we conclude (Q MID) and/or (PRLL)?” Write this out in words. This is a precise version of question (1) stated before Theorem 7. Can you answer it?