 In this video, I want to define the idea of supplementary angles in an ordered geometry. So imagine we have four points A, B, C, D that belong to an ordered geometry. Then we can say that the angles A, B, C and C, B, D are supplementary if the point B is between A and B, which be aware because of the axioms of betweenness. If B is between A and D, excuse me, then that actually means that A, D and B must be co-linear. And so basically with the following pictures what we have in mind when we think of supplementary angles here, we get a picture that looks something like the following. Here is our point B which is the common vertex to both of these angles, A, B, C and C, B, D. We have point A, we have point C, and we have point D. So we would say that the angle A, B, C and the angle B, D, excuse me, C, B, D, these are supplementary angles because the points A, B, D are co-linear with each other. And so we can say these angles are supplements of one another. And so that this gives us this idea of a supplement. So we should note that intuitively a supplementary angle means that the two angles when you add them together, you get a flat angle, that is you get a whole half plane like so, because A, B, C, A, B, D fall on the same line. So we have this notion of a supplementary angle. I should mention that given any angle, because angles make sense in any order geometry, given any angle A, B, C, let me erase some of my picture here. In fact, we'll just start from scratch. Given any angle, we'll draw it like this here for which this is A, this is B, and this is C. Given any angle in an order geometry, we can always construct a supplementary angle. Because after all, by the extension axiom, we can extend the line, right, A, B, so that there's a point D, some point D, such that B is between A and D. So the extension axiom gives us that. And therefore, because A, B, D are collinear with B in the middle, we're going to get that angle A, B, C is going to be supplement to angle D, B, C, right? So the existence of supplementary angles in a, in a order to geometry comes exactly from the extension axiom. All right, now we're going to transition a little bit. So we're going to go from an order geometry now to a congruence geometry, because in order geometry, we can't in general talk about what does it mean to be congruent to anything. That's exactly what we need. We need to be in a congruence geometry to talk about those type of things. So in a congruence geometry, we can actually say the following, supplements of congruent angles are congruent. So if we have two congruent angles, their supplements have to be congruent as well. And so what we're going to do is try to envision that exact situation right here. Imagine we have two angles, ABC, which are congruent to each other. And so you're going to see me drawing this picture right here. So we have these two angles for which one of the angles is given here. Here's angle A, excuse me, point A, point B, point C. Then we're going to have our other angle. This is A prime. This is B prime. And this is C prime right here. And by assumption, we're going to assume that angle ABC is congruent to angle A prime, B prime, C prime. So what we need to do is prove that the supplements of these two angles are likewise congruent. Well, by the extension action, which we just talked about, there do exist a point D such that the angle DBC is supplementary to ABC. And we also have a point, by extension, there's some point over here, D prime, so that the angle D prime, B prime, C prime is the supplement to A prime, B prime, C prime. So we need to argue, so our goal right now, so we want to prove that angle D, DBC is congruent to D prime, B prime, C prime, like so. All right, so how are we going to do that? So one trick that we're going to do all the time when it comes to these type of arguments is segment translation. This is something we do a lot because in essence, it allows us to move things around. So we have our points A, B, and C leave them alone. But if you look at the segment, if we look at the segment AB, we can translate this segment onto the ray B prime, B prime, A prime. And so there's going to exist some point over here, A double prime, such that, oh, this is too messy right now. Let me erase it real quick. So by segment translation, there exists some point A double prime on the ray B prime, A prime, so that the segment B prime, A double prime is congruent to the segment BA. Segment translation gives us such a thing. Now, it's also important to note that if you look at the angle A prime, B prime, C prime, this is equal to the angle A double prime, B prime, C prime. Because after all, when it comes to an angle, the exact location of the point doesn't matter because if you interchange, if we swapped A prime and A double prime, those are still on the same ray emanating from B and that just gives us the direction of the ray and the rays are what determines the angle there. So in other words, without the loss of generality, we can assume that A double prime is A prime because if it's not, we'll just relabel it and we use A double prime instead of A prime. So what I'm trying to say here is we can, without the loss of generality, assume, we can assume that the line segment, A B and A prime, B prime are congruent to each other because of segment translation. In other words, if A prime was too far or too short, we can scooch it in so that it's the same length, so to speak. Now, we don't have a notion of distance, we don't have a notion of length, but the same principle we can get from segment translation without the loss of generality because of segment translation, we can assume that the segment A B is congruent to A prime, B prime. Now, by similar reasoning, we can also assume without the loss of generality that the line segment B C is congruent to B prime, C prime. So we can assume these ones are as well, because after all, we can translate the segment B C onto the ray, B prime, C prime, that gives us some point C double prime, and we can just swap out C prime with C double prime so up to relabeling their congruent. And then likewise, same thing here, we can assume without the loss of generality because of segment translation that the segment B D is congruent to B prime, D prime, because after all, we can translate B D down here, we'll get some point D double prime. If that's not D prime, we just remove D prime and use D double prime instead, because I don't want to say double prime all the time, we'll just assume I was right the first time and we had D prime like so. So we can assume in addition to these angles, we have these line segments are congruent to each other. So segment translation is very powerful in that regard without the loss of generality, we can assume these things. So next what we're going to consider is we're going to form a triangle here. Let's consider the triangle ABC and the triangle A prime B prime C prime. I want you to notice here that under this assumption, we know that well under under the setup we have right here, A B is congruent to A prime B prime as a segment. We of course have the angle ABC is congruent to the angle A prime B prime C prime. And we have that the side length BC is congruent to C prime B prime as illustrated right here. And so these are exactly the assumptions of a side angle side triangle congruence. So by the side angle side axia, we actually can conclude that ABC the triangle is congruent to the triangle A prime B prime C prime as well. Now because corresponding parts of congruent triangles are congruent, we then can make some inferences because these triangles are now congruent to each other. So in particular, the side length AC that's going to be congruent to A prime C prime. Now segment translation couldn't give us that AC was congruent to A prime C prime. Moving around the points could mess up that congruence, but because of the side angle side axiom we get that. We also have that the angle BAC, which is this angle right here, it's going to be congruent to the angle B prime A prime C prime. So those angles are congruent to each other. Now it is also true that the angle ACB is congruent to A prime C prime B prime, but we don't need that observation. So I'm not going to make it at this moment. I want to keep the picture on the screen a little bit longer. So I'm actually going to enlarge it so we can still see it, but then we can go down the argument a little bit longer. So next by segment addition, we have that AD is congruent to A prime D prime. We could have done that earlier, but notice that this segment is congruent to this segment, and this segment is congruent to that segment. So by angle addition, we didn't know that the segment AD is congruent to A prime D prime. So we should probably make some mark of that. All right. So these ones are congruent. That's going to get messy though. So maybe we'll just highlight it here on the screen. We have that AD is congruent to A prime D prime, like so. And so then what we're going to do is form yet another triangle. Look at the triangle, the triangle AC, excuse me, the triangle ADC, and the triangle A prime D prime C prime. So if we look at what's happening there, we have that the segment AC is congruent to A prime C prime. We have that the angle CAB is congruent to the angle C prime A prime B prime. Now as you mentioned, that the angle A, excuse me, angle CAB is the same thing as the angle CAD, right? So this angle inside of the smaller triangle is actually an angle in the larger triangle. And then we also, the same thing we said down here, the angle C prime A prime B prime, of course, is the same as C prime A prime D prime. And then finally, by angle addition, we have that AD is congruent to A prime D prime. So therefore, the triangle ACD is congruent to the triangle A prime D prime C prime, like so. That's again an application of the side angle side right there. So ACD is congruent to A prime C prime D prime because of this side angle side situation. So this triangle is congruent to this triangle right here that I've now colored here in white. All right, so we've done two triangle congruences so far. Now because these triangles are congruent to each other, we have that their corresponding parts are congruent as well. So what's left to be done here? So I want you to look at these triangles here. So this segment, excuse me, CD is congruent to the segment D prime C prime because corresponding parts are congruent. We also have that angle BDC is congruent to angle B prime D prime C prime. Again, corresponding parts of congruent triangles are congruent. And we already knew from beforehand that the segment BD is congruent to the segment B prime D prime. So one final application of side angle side comes into play here. We have side angle side. This is going to give us that the triangle B prime D prime C prime is congruent to the triangle BDC, like so. These triangles are congruent to one another, again by side angle side. And so by corresponding parts, it would have to be that the angle BDC is congruent to the angle D prime B prime C prime that we talked about before. I'm going to zoom out a little bit so we can see the rest of the proof. I've illustrated, I've said everything that's going on here, but you can see that I need to move the picture down a little bit. Let me zoom in on the text for you and so you can read along or copy down for your notes if you want to. I just copied down the picture from above. Sorry about that. The magic of editing has made it all better right now. But we see these two triangles right here for which we took three applications of side angle side. We showed that the triangles ABC were congruent to A prime B prime C prime, then by side angle side, then using corresponding parts we are able to argue that triangle ACD was congruent to triangle A prime C prime D prime, again using side angle side, then using corresponding parts we found a few more congruences of angles and sides. We then were able to conclude that the triangle BDC was congruent to the triangle B prime D prime C prime, of course by side angle side. And then as corresponding parts are congruent, we get that angle CBD is congruent to angle C prime B prime D prime, thus showing that the supplements of the supplements of congruent angles are in fact congruent.