Proving triangle congruence | Congruence | High school geometry | Khan Academy

What I would like to do in this video is to see if we can prove that triangle DCA is congruent to triangle BAC.

Pause this video and see if you can figure that out on your own.

All right, now let's work through this together. So let's see what we can figure out. We see that segment DC is parallel to segment AB; that's what these little arrows tell us. You can view this segment AC as something of a transversal across those parallel lines. We know that alternate interior angles would be congruent. So we know, for example, that the measure of this angle is the same as the measure of this angle, or those angles are congruent.

We also know that both of these triangles, both triangle DCA and triangle BAC, they share this side, which by reflexivity is going to be congruent to itself. So in both triangles, we have an angle and a side that are congruent. But can we figure out anything else?

Well, you might be tempted to make a similar argument, thinking that this is parallel to that because it looks parallel, but you can't make that assumption just based on how it looks. If you did know that, then you would be able to make some other assumptions about some other angles here and maybe prove congruency. But it turns out, given the information that we have, we can't just assume that because something looks parallel or because something looks congruent, that they are.

So based on just the information given, we actually can't prove congruency. Now, let me ask you a slightly different question. Let's say that we did give you a little bit more information. Let's say we told you that the measure of this angle right over here is 31 degrees and the measure of this angle right over here is 31 degrees. Can you now prove that triangle DCA is congruent to triangle BAC?

So let's see what we can deduce now. Well, we know that AC is in both triangles, so it's going to be congruent to itself. And let me write that down. We know that segment AC is congruent to segment AC. It sits in both triangles, and this is by reflexivity, which is a fancy way of saying that something is going to be congruent to itself.

Now, we also see that AB is parallel to DC, just like before, and AC can be viewed as part of a transversal. So we can deduce that angle CAB—let me write this down, actually doing a different color—we can deduce that angle CAB is congruent to angle ACD because they are alternate interior angles where a transversal intersects two parallel lines.

So just to be clear, this angle, which is CAB, is congruent to this angle, which is ACD. And so now we have two angles and a side, two angles and a side that are congruent. So we can now deduce by the angle-angle-side postulate that the triangles are indeed congruent.

So we now know that triangle DCA is indeed congruent to triangle BAC because of angle-angle-side congruency, which we've talked about in previous videos. And just to be clear, sometimes people like the two-column proofs. I can make this look a little bit more like a two-column proof by saying these are my statements, and this is my rationale right over here.

And we're done.