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
A flowchart proof shows one statement followed by another, where the latter is a fact that is proven by the former statement.
Recall the isosceles triangle theorem: two legs are congruent, then the two base angles must be congruent. The converse of this is that if the two base angles are congruent, then the legs are congruent. To start the flowchart proof, the first statement is what is given in the problem--the fact that the base angles are congruent and that there is an angle bisector that splits the triangle into two triangles. Then, state that these triangles are congruent by AAS (the angles are the base angle and the bisected angle and the side is the angle bisector, which is congruent to itself). As a result, the legs of the triangle are congruent.
In this problem we’re going to use a Flowchart Proof to prove the converse of the isosceles triangle conjecture. So we’re being asked to show that if the two base angles are congruent then this side, AB and this leg BC, must be congruent. So that’s the converse of the isosceles triangle conjecture. So let’s start off by marking our diagram.
Well I see that angle A and angle C are congruent, so I’m going to mark these two angles as congruent. I also see that line segment BD is an angle bisector, which means this line segment right here creates two congruent angles. Now a strategy you can use in Flowchart Proofs is working backwards.
So what I’m going to do is I’m going to start by writing my last step which is triangle ABC is isosceles. So we go all the way to the end board and write the last step of our proof, which says triangle ABC is isosceles, and our reason is going to be definition of isosceles triangle. I’m just going to abbreviate triangle with a little symbol.
Now in order to say that we need to ask ourselves, what is the definition of isosceles triangle? Well if I sketch it right here, an isosceles triangle has two legs that are congruent. So we’re going to need to say that these two legs are congruent. So let’s go back to our diagram here and I see that I’m going to have to show that AB is congruent to BC.
So let’s go back and write that, that line segment AB is congruent to line segment BC. Now the only way we’ll be able to say this is if we have two corresponding congruent triangles. So the reason here is going to be CPCTC. In order to say that two triangles are congruent and corresponding we actually have to have two triangles that are congruent. So I’m going to write that triangle blank is congruent to triangle blank for some reason that we need to figure out.
So let’s go back to our diagram and if I just look at these, we have two pairs of corresponding angles that are congruent, not enough information. But if you look closely you see that they share a common side, BD. So BD is going to be congruent to itself. So we can use the angle-angle-side (AAS) short-cut to show that these two triangles are congruent. So we can go over here and we can say angle-angle-side (AAS) is our short-cut. Again we’re just working backwards.
So now let’s figure out what are the corresponding angles? Well I see that angle A and angle C both have one congruency marking, so those are going to be my first letters. I’m going to say angle A and C, so now we need our second letters. Well, let’s go with angle B which is going to correspond to itself. So B will be our second letter. So far we have triangle AB is congruent to triangle CB. We need our one final letter here and we see that this angle D is going to correspond to itself as well. So we have triangle ABD is congruent to triangle CBD.
So now in order to say this, we need three pieces of information. We need to show that those two angles are congruent and that those two sides are congruent. So let’s start of with what we are given. Well we know that these two angles, A and C are congruent. So I’m going to write angle A congruent to angle C and my reason here is given. Let’s go and say line segment BD is an angle bisector. Well does that actually tell us that we have two congruent angles? No. So I’m going to say that angle ABD, so I’m going to write that over here, angle ABD is congruent to the other angle here, CBD. So we have angle CBD and we’re going to say that this is true because of the definition of an angle bisector.
Now in order to say this, we actually have to start it off by saying BD is an angle bisector. So that’s going to be my statement up here, that line segment BD is an angle bisector. And that is given. So notice that we actually had to take a step back from our three statements about congruence.
So our final statement is the two sides that are congruent and that’s BD congruent to itself. So I’m going to say line segment BD is congruent to line segment BD and our reason there is reflexive property, which means it’s congruent to itself.
So notice what we did, we said it’s okay to work backward from the proof and start where you know you need to finish. And we also realized that this was not enough information because we had to show why we knew it was an angle bisector. Those are the two keys to coming up with a great Flowchart Proof.