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Two Column Proofs - Problem 3

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

Given two overlapping triangles, prove the congruence of two angles by first marking which segments of the triangles are congruent. The first statements are the given congruences. Then, both angles have a congruent angle since they are overlapping and thus share an angle. By examining the congruence of the sides and angle, the triangles are congruent by SAS. By CPCTC, the corresponding angles are then congruent.

In the proof, statements are facts and reasons are the theorems or properties that cause the statement to be true (such as SAS or CPCTC).

Another common proof is when you have overlapping triangles. So notice that we have two small triangles here, so we’ve got these two little guys and then we also have two larger triangles. So the trick to this question is going to be deciding are you going to show that these smaller triangles are congruent, or you’re going to show that the two larger triangles are congruent?

Well let’s start by marking what we’re given and that might help us figure it out. We know that BD and AD are congruent. Well that’s going to be these two big sides, so what I’m going to do is I’m just going to mark, I’m going to redraw that side so I remind myself that these two are congruent.

The second thing that they gives us is that ED and CD are congruent to each other, so I’m going to mark that these two sides are congruent.

Now the trick to this problem is realizing that angle D is congruent to itself, so I can see that if I look at these smaller triangles, I don’t have enough information to say that they must be congruent. So we’re going to have to use our larger triangles.

The next key trick to this is to redraw these triangles because right now it’s kind of a mess. So I’m going to redraw this triangle ACD and I’m going to mark it. We said that AD has one marking. We had CD with two markings and we know that angle D will be congruent to itself. So let’s redraw the triangle BDC, BDE excuse me.

So I’m going to redraw that right here and we see that A and B are going to correspond because we know it’s what we’re trying to prove. We see that angle E and C are going to correspond and angle D is going to be corresponding to itself. So angle D is going to have one marking, BD will have one congruency marking for side, and ED will have two markings.

So now we ask ourselves do we have enough information to say that these two are congruent? Yes we do we have side, an included angle and another side. So our reason is going to be side-angle-side. If these two triangles aren’t congruent, then we can end up saying that A and B must also be congruent. So let’s go over and start out two column proof.

Our first three statements are going to be same to two sides, two angles and two sides are congruent. Let’s start with our given. We said that BD is congruent to AD and our reason is given. Our second statement is going to be that the two angles are congruent, so we’re going to say angle D is congruent to angle D and the reason is it’s the same angle so it’s reflexive, so I’m going to say reflexive property. My third statement is going to be this other corresponding side, which is ED is congruent to CD and our reason is given.

So now we have enough information to say that two triangles must be congruent. So I’m going to start by saying triangle ACD, and this order doesn’t matter but it sets your order for how you’re going to list those vertices in the second triangle, so I’m going to say it’s congruent to triangle A corresponds to B. We know that D is going to correspond to itself, so I’m going to write D last, which means E has to be the middle letter and our reason for this we’ve already written that down that is side-angle-side.

And our last statement now we said that two triangles are congruent, is that corresponding parts must be congruent, so I’m going to say that angle A is congruent to angle B and our reason, that tongue twister, CPCTC.

So the two keys to this problem: one was realizing that angle D would be congruent to itself and the second thing was determining which triangle are we going to use and we determined that by saying we have enough information to say that those two big triangles must be congruent.

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