 Welcome back to our lecture series, Math 3130, Modern Geometry for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In lecture 18, we are going to talk about various triangle congruences. Now, I want to draw your attention to the one triangle congruence we have. So as an axiom, the sixth axiom of congruence, we have taken upon us the side angle side axiom. What we're going to do in lecture 18 is develop other triangle congruences. Now, this side angle side has been quite trusty for us so far, but we're going to see in this video that there's also an angle side angle congruence. That is, if two angles are congruent and the side length between them is also congruent between two triangles, the two triangles would then be congruent to each other as well. In the next video, we're going to prove that the side side side triangle congruence is valid in congruent congruence geometry, and we're also going to prove the side angle angle triangle congruence. So I'll have these four different triangle congruences in congruence geometry. All right. I should mention though in this video that there are of course limitations to triangle congruences. For example, there is not going to be an angle angle angle triangle congruence in congruence geometry. What that means is of course is that if you had two triangles which have the same three angles up to congruence, that doesn't necessarily guarantee that you have congruent triangles. In fact, in Euclidean geometry, angle angle angle doesn't imply congruence at triangles at all. It does give some notion of equivalence, not the equivalence of congruence, but it does give us the notion of similar triangles. Triangles that are proportional to each other. Their sides are proportional to each other, but they're not necessarily congruent to each other. And we see arguments like this all the time when we do stuff in Euclidean geometry. So like as we see like water that fills up in some type of triangular tank or something, these triangles are going to be proportional to each other. They're going to be similar to each other. Their angles are the same, but the side lengths do change throughout the whole thing. Now interesting enough, in hyperbolic geometry, angle angle angle actually is a congruence condition. That is if two hyperbolic triangles have the same three angle measurements, then it turns out that those two triangles are congruent to each other. Elliptic geometry has this exact same property. So wall angle angle angle is not going to be a triangle criterion for neutral or congruence geometry. There are some geometries where it is one, but it won't be the case in congruence geometry because it's not valid in Euclidean geometry, which is a congruent geometry. I also want to mention the situation of side side angle. This is the one that's played around a lot in, for example, trigonometry, math 1060 here at Southern Utah University. This is sometimes referred to as the ambiguous case because in Euclidean geometry, side side angle doesn't guarantee that two triangles are congruent to each other because what you have here is you have an angle as congruent. You have two sides that are congruent, but the angle is not between the two sides and play here. You actually get a couple possibilities that if you have the side side angle situation, it could be that there's no, the two triangles aren't congruent to each other. It could be that they are congruent to each other. In particular, you could have two different incongruent triangles that have the same side side angle measurements. That's why in Euclidean geometry, it's called the ambiguous case. These same problems do also exist in hyperbolic and Olympic geometry as well. So the side side angle situation, we can't really do much with it. There is a special case in right triangles, the so-called hypotenuse leg condition that we'll talk about some other time, not of course in this lecture right now. So let's actually get to the main topic for this lecture. We want to prove that in a congruence geometry, that angle side angle is a congruent triangle condition. That is, if you have two triangles with two congruent angles and the side between them is congruent, then the two triangles have to be congruent to each other. So let's actually draw a picture to be explicit what we mean here. So we have one triangle, we're going to call that triangle ABC, and so we'll just label it like this, ABC. And then we have some other triangle. I'm going to do it in red over here. And don't worry if I don't make my triangles look the same, that's okay. I'm not actually supposing that they are congruent to each other yet, but we'll get to that eventually. So we have two triangles, ABC and triangle A prime, B prime, C prime, but we do have some assumptions of congruence. So we're going to assume that the angle BAC is congruent to the angle B prime, A prime, C prime, like so. We're going to assume that the line segment AB is congruent to the line segment A prime, B prime, and we're going to assume that the angle ABC is congruent to the angle A prime, B prime, C prime. So in this situation, we have the angle side angle situation. We then claim that these two triangles are going to be congruent to each other, thus proving the angle side angle criterion. Now, because the only criteria we have so far is side angle side, we're going to have to somehow turn angle side angle situation into a side angle side situation. And so that's what our plan is going to be right now. By segment translation, there's going to exist some point C double prime. And in fact, this is a unique point on the ray, on the ray A prime, C prime, such that the segment AC is congruent to the segment A prime, C double prime. So what I want to show you here is that we can extend the segment A prime, C prime, and so there's some new point here, C double prime, such that the segment AC is congruent to the segment A prime, C prime, A prime, C double prime, excuse me. And so then we want to consider the triangle A prime, B prime, C double prime, and compare that to the triangle ABC. So you'll notice that in this situation that when you look at the triangle ABC, we have a congruent side with the triangle A prime, C double prime, we have a congruent angle, so angle A and angle A prime are still congruent. And by assumption, the segment AB was congruent to the segment A prime, B prime. So therefore, by side angle side, that was how we made the connection there, by side angle side, the triangle ABC is congruent to the triangle A prime, B prime, C double prime. So these triangles are congruent to each other, ABC with A prime, B prime, C double prime. So in particular, because corresponding parts of congruence triangles are congruent, we get that the angle ABC, which is this angle right here, is congruent to the angle A prime, B prime, C double prime. Now by transitivity, since the angle A prime, B prime, C double prime is congruent to A prime, B prime, C prime, as they're congruent to each other because they're both congruent to angle B over here. Then of course, by uniqueness of angle translation, notice that where our angles live, we have the ray B prime, A prime, like so, both C and C prime are on the same side of the line A prime, B prime. And so by uniqueness of angle translation, it's got to be that these angles are actually one and the same thing. So the point C prime and the point C double prime are actually the same, the same points. It says here, of course, that the rays are equal, which that was always the case. Really, what we should be saying is that the points, the point C prime and the point C double prime are equal to each other. Because after all, that was a unique point that satisfied that segment translation. So by uniqueness there, oh, excuse me, I did have this right the first time. Sorry, we're looking at the ray B prime, C double prime, and the ray B prime, C double prime. So I'm sorry, these two rays are in fact equal to each other, which is then going to force that C and C double primary, C prime and C double primary equal to each other. So that was all correct as well. And since C prime is equal to C double prime, this actually shows that the triangle A prime B prime C prime is equal to the triangle A prime B prime C double prime. And as this was congruent to ABC, then we know that the triangles ABC are congruent to A prime B prime C prime, thus proving the angle side angle situation. So I want to summarize what we've done so far. In our congruence geometry, we assume side angle side as our axiom of triangle congruence. We then have proven that angle side angle is a theorem of congruence geometry. And essentially the proof was side angle side. Now, yes, we did use some segment and angle translations and things like that, transitivity of congruence. The other axioms are on play there. But the main argument was we turned an angle side angle situation into a side angle side situation, got triangle congruence, and then we then had to determine that by changing the problem, it really was still the same triangle in the end. No big deal there. Now I want to point out that this is really just a matter of flavor that if one wanted to, you could have actually assumed angle side angle as your axiom of triangle congruence. And by a similar argument, we could improve side angle side as a theorem of angle side angle by changing the appropriate parts of this proof we just did right here. That's an exercise I will leave to the viewer. And in the next video, we're going to prove the side side side triangle congruence.