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.
To prove that two segments in a trapezoid are congruent, first show that the triangles formed by the diagonals are congruent. The first statements should be regarding the congruence of angles or segments, as given in the problem. Then, based on these statements, the congruence of the triangles can be proven by SSS, SAS, ASA or AAS. The final statement is that the segments are congruent by CPCTC.
An Application of Flowchart Proofs is a problem like this where we have overlapping triangles and triangles within triangles. We’re given a couple of key pieces of information. We’re being asked to prove that AC and BD are congruent. So first let’s find those segments.
AC is that Diagonal and BD is that diagonal, which automatically eliminates this small triangle, AED and this small triangle, BEC. Because if you notice this triangle right here only contains part of segment AC. So by proving these two triangles congruent, that won’t help us at all. What we are going to do is we’re going to prove that these two larger triangles must be congruent.
So let’s start by marking this. Angle ADC, so we have ADC, I’m going to mark this with one line, is congruent to BCD, so BCD is that big angle. Second key piece of information is that line segment AD is congruent to line segment BC. So we have line segment AD congruent to line segment BC. But this is a little confusing.
So when you have a problem like this I suggest that you redraw your triangles. So I’m going to redraw these triangles in a way that it will be pretty easy to see corresponding vertices. So I’m going to make these triangles look pretty much identical and I’m going to start with my triangle ADC. So I now I need to mark it. Where I see AD has one marking, I know that this angle D is going to be corresponding and congruent to angle C and that’s all I know for now.
So let’s write our other triangle. We know that B and A must correspond because they are both up at the top there. We know that angle C is going to be down here, because we know that C and D must be congruent, and out last vertex is D. So here is a trick that we’ve used throughout geometry and that’s saying that any line segment must be congruent to itself. So I’m going to mark that DC is congruent to CD and that reason is going to be reflexive property.
So notice that we have enough information now. We have side-angle-side (SAS). So the game plan is to say that two sides, two angles and two other sides are congruent, which means these two triangles are congruent, which means we can say these two segments are congruent.
So let’s go over here and start our Paragraph Proof and actually we’re not going to do a Paragraph Proof, we’re going to do a Flowchart Proof. So let’s start with our first statement.
Our first statement is our given, angle ADC is congruent to angle BCD, so I’m going to write that in here. Angle ACD is congruent to angle BCD and then outside of the box is going to be our reason. And the reason here is given.
Our second statement is going to be the two angles that are congruent. Actually we just said the two angles. So we need our two sides now. So I’m going to say that AD and BC are congruent. So we have line segment AD congruent to line segment BC and I’m going to draw that in a box. I’m just saying that is our given. And last we need these two sides that were congruent to themselves. So that’s DC congruent to DC, and our reason is reflexive property of congruence, and I’m just going to write reflexive property.
So these three reasons allow us to say that the two triangles are congruent. So I’m going to draw in some arrows in order to say the two triangles are congruent. So we're going to say triangle and I’m going to come back now to my original drawing, we have ADC, so I’m going to write that, right over here that triangle ADC is congruent to, well, what corresponds to A? To do that, we’re going to have to go back to our drawing. Angle A corresponds with angle B. So now I have to walk all the way back and say that angles A and B are corresponding and congruent. Well, what corresponds with angle D? So let’s go back. Angle D we said is corresponding and congruent with angle C. So I’m going to walk back over here and say that angle D and angle C are corresponding and congruent. Which just leaves our last vertex; angle C corresponds to angle D.
Now the reason why we have to do this is because this is a very specific order that if you did it wrong you’re going to get the problem wrong. So I’m going to draw a box around this and notice what we said, we said that we’ve got and angle, two sides and we said our short-cut was side-angle-side (SAS).
Now here comes the easiest part of the proof and that is we’re just going to rewrite what we’re being asked to show. So here we’re being asked to prove AC congruent to BD. So I’m going to write this is in my last box. AC congruent to BD. Line segment AC is congruent to line segment BD and our reason, because we have two corresponding and congruent triangles, is CPCTC. So notice what we started with our three reasons of saying two triangles are congruent. We said they’re congruent and why and then we we’re able to make our final statement of proof.