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Two Column Proofs - Problem 2

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

When given a parallelogram with one diagonal, it is possible to prove that two angles in opposite corners are congruent. Since the polygon is given to be a parallelogram, it is known that there are two pairs of parallel sides. Additionally, the parallel sides are congruent. Additionally, the diagonal is a shared side of two triangles, so, it is possible to prove that the triangles are congruent by SSS. Since corresponding parts of congruent triangles are congruent, the corresponding angles (for example, the ones in opposite corners of the parallelogram) are congruent.

I can almost guarantee that you’re going to see a problem like this where you have a quadrilateral and two triangles within it and you’re being asked to show that two angles or two sides are congruent. So let’s start off by marking our figure since we don’t have any markings on it right now.

We’re given 2 key things. The first is that AB and CD are congruent, so I’m going to mark those two lines as congruent.

The second key thing is that BC and AD are congruent. Notice that I’m using different number of markings here because we don’t know that this is a rhombus or a square. We’re being asked to show that angle A and angle C are congruent. So to do this I’m going to redraw this triangle CBD. I’m going to redraw it right over here, so that way it’s a lot easier to compare the two triangles. Well looking at this AB has one marking CD has one marking, so I know that C is going to correspond to angle A and that D is going to correspond to angle B, which leaves our last angle down here is B.

So let’s transfer these drawings over. We’ve got CD with one marking and we’ve got BC with two markings. Now this is not enough information to say that these two triangles must be congruent. So the third side, BD, is in both of these triangles. It’s the same segment so it has to be congruent to itself. So let’s move over to our statement and reason our two column proof.

Well to start off we need to know where we’re going. We’re going to prove that these two triangles are congruent by side-side-side. Once we’ve said that then we can say that if they are congruent, then all the matching parts or corresponding parts must also be congruent. So that’s what we’re going to end with, that angle A is congruent to angle C and we’re going to say by CPCTC that tongue twister.

So first we have to show that these three sides are congruent. Let’s start off with out two givens that’s the easiest place to start so I’m going to say that AB line segment is congruent to line segment CD and our reason is given. Usually you’re going to have at least one or two things that are given.

Our second statement is our other given and that is BC is congruent to AD and our reason is given. Now our third side, so we’re done with two of these sides. Our third side is this DB. So I’m going to write DB is congruent to DB, so that’s your hint that we’re going to use what’s called the reflexive property which means it has to be congruent to itself, so I’m going to write reflexive property.

Our fourth statement has to be that these two triangles are congruent. I can’t say that corresponding parts are congruent without first saying that the two triangles are congruent. So I'm going to say that triangle ABD, and that order doesn’t matter, is congruent to triangle, and this order matters because A corresponds to C. So now I’m happy that I redrew my triangle here because it’s really easy to pick out my corresponding angles. So we have C, angle B, in that triangle, corresponds to angle D and our last is D corresponds to B.

So we’ve got our two triangles that are congruent and if you mess up this order you’re definitely going to lose points on your test or quiz and our reason we said was side-side-side. And now comes the easiest part of any proof and that’s just rewriting what you’re being asked to show.

So we can conclude that angle A must be congruent to angle C and our reason is, if these two triangles are congruent then all of the matching parts are also congruent, so we’re going to say CPCTC. So the two keys here one was redrawing our triangle and the second key was knowing that corresponding parts of congruent triangles must be congruent.

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