 Alright, here we have our final example for the day. We've got a goal of proving that MN and EA are congruent. Now in order to do that, we're going to have to prove that the two triangles are congruent. So those two triangles, let's name them. So once again, proving that these triangles are congruent is going to be kind of a subgoal in order to prove that these segments are congruent using the CPCTC. So let's piece this together. Which of our triangle congruence theorems can we use? We know that MN is parallel to EA. So MN and EA are parallel. And what that tells me is any angles that are created by those segments will be congruent, either congruent or supplementary. So if we consider MN and EA and they're cut by that transversal, then the angles that created them, sorry not that created them, but the angles that are created, if we look down at these angles, we've got angle two and angle M would be considered corresponding angles. So those are congruent. So that fact tells me that angle two is congruent to angle M. We have given information that N is congruent to A, so this angle and that angle are congruent. And then lastly E is the midpoint of MG. Well what does midpoint mean? Midpoint is the middle point. So here's MG and E is right in the middle. In other words, that segment, ME, has to be congruent to EG. So now in these triangles, let's just pull out one of those triangles. In these triangles, we are given that these blue angles are congruent. So that's angle M and angle two. We're given that these green angles are congruent, angle N, angle A, and then finally we've got that midpoint segment, or that midpoint of a segment which means that these segments are congruent. So we've got the green, the blue, and the red. We can either talk about angle angle side or angle side angle. And how do we pick? Well, in angle side angle, this side must be the included side. In this triangle, this side, this side with the red tick marks is not the included side because it only touches that blue angle. It has nothing to do with this green angle over here. So as a result, we're not going to use angle side angle. We must use angle angle side in order to prove that the triangles are congruent. So let's get started. We know we're, since we're proving that segments are congruent, we must first prove that the triangles are congruent. And then once we know all the triangles are congruent, from that information, we can then prove what it is we're trying to prove, specifically that MN is congruent to EA. So hopefully these last two steps, let's call this like step one, step two. Hopefully these last two steps are looking familiar to you. First proving that the triangles are congruent. And then second using that congruence to prove that the segments or the angles or some parts of the triangles are congruent. All right, now since we're using the angle angle side theorem, we need to talk about which pair of angles, which pair of other angles, and then which pair of sides are congruent. Well, let's, we can just start in order of the given information. We know that MN is parallel to EA. So MN is parallel to EA. And we saw that that gave us these corresponding angles are congruent. So angle two and angle M are congruent. So parallel lines make corresponding angles congruent. Next, we could take a look at the next piece of given information, says angle N and angle A are congruent. And so that can go in our next box. And we need no statement above that box because that was given information. So we don't need to explain why it is because, well, it's just given to us. And then finally, the segments, we knew that ME and EG were congruent. And the reason was, was E is the midpoint of MG. And midpoint means, midpoint, middle point, it's splitting up the segment into two parts, two equal parts. And so that midpoint definition will give us that information. All right, and now I think we're done with this proof. All of our given information is used up. Here we've got the green is used, the blue is used, and the red is used. And we've proven first that the triangles are congruent and then using the CPCTC that the segments are congruent. So we're all done.