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Write a two-column proof.

(FIGURE CANNOT COPY)

Given: $\overline{J K} \cong \overline{N M}, \overline{K P} \cong \overline{M Q}, J Q>N P$

Prove: $\mathrm{m} \angle K>\mathrm{m} \angle M$

$\angle K>\angle M$ by the converse of the hinge theorem.

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Johns Hopkins University

Piedmont College

Oregon State University

University of Michigan - Ann Arbor

we've been given a pair of overlapping triangles and we're trying to prove that the measurement of Angle K is greater than the measurement of angle em. So we want to start with our given statements of segment JK, which is on our left triangle, is congruent to segment M and which is on our right triangle. Also segment KP, which is on our left triangle, his country congruent to segment MQ, which is on our right triangle. These air given to us as givens And then they also tell us that segment the length of segment Jake you is greater than the length of segment and P, and this is gonna become very important when we look at the basis of our two triangles. So this is a given. So one of the pieces of the triangle that's missing is this segment. Q p. Well, it's in both triangles, so I'm gonna say segment Q P is congruent to segment Q P. And this is by reflexive property abbreviate of equality. And I'm going to add that segment to each of our other based segments. So we've got Jake. You plus Q P is equal to J. P and I'm gonna use NP plus Q P is equal to an que and both of these our segment addition cost Hewlett. So now that we have the third side of each triangle, we want to go back to this idea of ah, segment Jake, you being greater than segment JN. So that means that segment Jake you plus Q P is gonna be greater than segment np plus Q P. On that is just the addition Property of equality because we added the same thing to both sides of our given inequality statement up here. Well, based on our segment edition postulate it, we can use substitution and say that Jake, you plus Q p in this statement is equal to JP and N P plus Q P, which we wrote in this statement here is equal to N. P. So we now have the third side of each triangle. However, one side is greater than the other. So we can compare the angles opposite of these sides and say that the measurement of anger okay, because it's opposite of the larger side, is gonna be a larger angle. Then the measurement of angle em and this is you're converse hint here. Um, and I left off the reason of the statement above it. And this was just a substitution property of equality because we use substitution.