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Two Column Proofs - Problem 1

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

To show that two triangles are congruent in a two column proof, first mark the diagram, if provided, using the given information about that triangle. This involves marking the segments that should be congruent. For example, remember that a midpoint divides a segment into two congruent pieces. Working backwards from the goal (which is to show that the triangles are congruent), notice which angles and sides are congruent and corresponding. Applying the SSS, SAS, ASA, AAS, or HL shortcut to these congruent/corresponding sides and angles, you can show that a triangle is congruent.

Let’s look at the two column proof, where you’re being asked to show that two triangles are congruent. Usually when you’re showing two triangles congruent, you don’t need to use that tongue twister, CPCTC. Let’s start off with what we’re given.

Always, if it’s not marked, always start by marking your diagram. First we know that line segment AB is congruent to line segment BC, so here we’ve got AB, I’m going to mark that congruent to BC. Now instantly I’m going to think this is an isosceles triangle because I have two lines that are congruent.

So I’m going to go ahead and I’m going to mark angle A and C congruent. Next it says D is a midpoint. Well here is point D, so that means that it bisects this line segment AC, so I’m going to mark AD and DC congruent, which is the definition of a midpoint. So now let’s go over and take a look at our two column proof.

We have one column for statement and one column for reason. So our last statement is going to be triangle ABD is congruent to CBD, so we have to work backwards. Let’s start off with how are we going to prove these two triangles congruent? I see that I have two sides that are congruent and corresponding. I have two angles and two more sides, so my short-cut is going to be side-angle-side. So my first three statements are going to be proving two sides congruent, two angles congruent and then two more sides.

So our first statement is going to be AB and BC are congruent, so I’m going to write that. Line segment AB is congruent to line segment BC and my reason, again I’m using number one for both of these, is that it was given.

My second statement is going to be about these angles. I’m going to say that angle A is congruent to angle C and my reason is going to be definition of isosceles triangle and I’m going to abbreviate definition, because we know that AB and BC are congruent that means that we have an isosceles triangle.

So my third statement is going to be about my other two sides that are congruent, so I’m going to say that AD is congruent to DC and my reason is going to be definition of midpoint. And again I’m going to abbreviate ‘definition’. So now we have 1, 2, 3 reasons to say that these two triangles must be congruent, which is going to lead me to my last statement, which is going to be the proof part.

So I’m going to say triangle ABD is congruent to triangle CBD and our reason we’ve already written it down, that’s side-angle-side. So we started backwards, we said well how are we going to proof that these two triangles are congruent? And then we showed our reasons. Remember, your last statement will always be what you’re being asked to prove or to show.

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