 So we can use the CPCTC to prove that triangles are congruent, right? So, so all we would have to do is show that all pairs of corresponding parts are congruent and then we could say the triangles are congruent. Well, the problem with that is we'd have to establish six different things before we could prove that the triangles are congruent. Not only would we need to prove that all three pairs of sides are congruent, but we'd also have to prove that all three pairs of angles are congruent as well, which is tedious. That's a whole lot of work. Well, the nice thing is we don't need to have all six things. In fact, we've got a couple of theorems that allow us to cut that number in half. So I'll give you a couple of examples in order to prove that triangles are congruent. One of the theorems that we have to prove that triangles are congruent is called the SSS theorem. Well, just a sec here. And what that means is if we can prove that these segments are congruent, that these segments are congruent, and that these segments are congruent, if we can prove that those three pairs of segments are congruent, then we can guarantee that those two triangles themselves are congruent. So that's called SSS, or side, side, side. Another theorem we have to prove that triangles are congruent is called the SAS theorem. Now, SAS stands for side angle side. And so what that means is we need two pairs of sides. Let's say these sides were congruent and these sides were congruent. And then we need an angle to be congruent as well. However, that angle must be the included angle. And what that means is the angle that we know something about must touch the two sides that we know about. So if we know that these two angles were congruent, then we could prove that the triangles themselves are congruent. However, if instead of those two angles, let's say we knew that these two angles were congruent, that is not enough to say that the triangles themselves are congruent, because that's not the included angle. So by included, I mean it has to touch the two sides that we know something about. Let me show you another example. If instead of these two sides, what if instead of those sides we knew that, let's say, that side, those pairs of sides were congruent, and these sides were congruent? Well, we could still use side angle side if we knew that these angles were congruent. Because again, they're the included angles. They're in between, or they sort of follow the sides that we know something about. So that's side angle side. The next theorem we have is called ASA, or angle side angle. This theorem is similar to side angle side, except now we're using the angles as, well, I guess you could kind of think about it like a sandwich. Let's say that we knew that these angles were congruent, and let's say these angles were congruent. If the included side was also congruent, the included pairs of sides, if those were congruent, then we could prove that these two triangles are congruent. So again, that side must be the included side. So another quick example with this one. If we knew that these two pairs of sides were congruent, and these segments were congruent, that would be enough to prove that the triangles themselves were congruent. And one final theorem that we'll deal with is called AAS, in other words, angle angle side. For this theorem, we want to make sure that the side we're talking about is not the included side. So an example of this, let's say we have these angles congruent, and we've established that these angles are congruent. Now it's possible that if these sides were congruent, then we would follow angle angle side. And again, it's kind of like the order in which you see the things. Here's angle angle side, angle angle side. So we have those four theorems that we're dealing with. Side side side, side angle side, angle side angle, and angle angle side.