 This next series of videos will deal with proofs using the ASA and the AAS triangle congruence theorems. So let's take a look at our first example. So we're given that WT bisects NP, and we want to prove that these two triangles are congruent. So first things first, we need to name the triangles. So if I start with angle N here as one of the sides of my triangle, the corresponding angle, well, in this second triangle, we have a choice. It's either going to be T or it's going to be P, and how will we know? Well, angle N is created by this long parallel side and then this long side connecting to E. And likewise, angle P is made from this parallel side and the long side connecting to E. So N corresponds with P. And from there, we can kind of piece together the rest. If we go from N to E, end up at W, then again we can start at P, head toward E, end up at T. So please also note that we have these arrows on segment PT, on segment PT and on segment NW. And those arrows indicate parallel lines. So that's other given information and W is parallel to PT. So how are we going to go about proving this? We need to prove that these two triangles are congruent and, well, we're missing some stuff. We can prove triangles are congruent using SSS, SAS, ASA, and WAS. And right now we have no sides congruent and no angles congruent. So let's piece a few things together. Let's take a look at this given statement, the fact that NT, sorry, WT bisects NP. So WT is doing the bisecting but NP is being bisected. NP refers to this long segment and we know that that segment is bisected. Or in other words, it's cut into two equal parts and WT is doing that cutting. Secondly, we have these parallel lines. And parallel lines, well, if we consider these parallel lines cut by the red transversal, then angle P and angle N would be considered alternate interior angles. And then finally, the way that these two triangles are drawn, we have intersecting lines or intersecting line segments and intersecting line segments create vertical angles. So we've got the blue angles, the red segments, and the green angles means that we can prove using the ASA there. So let's get to it. Let's write our proof. So right away, we know that the given statements, well, the given statements are there, but the prove statement, that's what we're trying to prove, so that goes in that last box down here. And in the previous slide, we ascertain that we're going to use the ASA theorem to prove that. And so we need to talk about which pair of angles, which pair of sides, and which pair of angles are congruent in order to prove that the triangles are congruent. Well, we've established that angle N is congruent to angle P. And the reason why is we have those parallel lines, N, W, and P, T. Next, we need to talk about the segments that we had congruent. We knew that N, E, and P, E were congruent. And that came from the fact that W, T bisect N, P. And then finally, we had our vertical angles. And those come from the fact that, well, we have intersecting lines because of the way that this particular figure is drawn. So those vertical angles are always congruent, which means angle N, E, W is congruent to angle P, E, T. And so there's our proof. We are done. Let's scroll down a little bit so you can see everything.