 Welcome back to our lecture series, SMATH 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misildine. In the previous lecture, lecture 14, of course, we introduced the congruent axioms relative to a line segment, and then we defined the idea of a congruent geometry one, which is an order geometry plus the six axioms of congruence. We mentioned the first three and I promised you that in the next lecture, which of course is now lecture 15, that we would introduce the notion, well, the congruence axioms for angles. The last three are very similar to the first three. As a reminder, we defined, well, I shouldn't say we defined, we introduced an undefined term for congruence of line segments, for which then we got segment translation, transitivity of segment congruence, and we had segment addition. In this video, we're going to introduce the idea of congruence between angles. So if we have two angles, ABC and DEF, we can say that two angles are congruent. It is an undefined term. And again, it's going to be a homonym to we talk about congruence of line segments. We're not going to say that an angle is congruent to a line segment. A segment can be congruent to another segment. An angle can be congruent to another angle. They are different things, but they are, of course, related. We will use the same symbol, of course, to represent congruence. We say angle ABC is congruent to DEF, and we'll use this symbol that we see here on the screen. All right? So with this notion of congruence of segments and congruence of angles, we can now define what it means for two triangles to be congruent, okay? So recall that a triangle is going to be, the triangle ABC is the intersection of the three angles, ABC, BCA, and CAB. I hope I listed all those correctly there. But a triangle, it's the intersection of the three angles associated to it. So suppose we have two triangles, triangle ABC and triangle A prime, B prime, C prime. So then we're going to say that triangles ABC and A prime, B prime, C prime are congruent if the line segment AB is congruent to A prime, B prime. The line segment BC is congruent to B prime, C prime. The line segment AC is congruent to A prime, C prime. The angle ABC is congruent to A prime, B prime, C prime. The angle BCA is congruent to angle B prime, C prime, A prime. And lastly, that the angle CAB is congruent to the angle C prime, A prime, B prime. So we say that two triangles are congruent to each other if they're three side, the line segments that form its three sides are congruent and the three angles are congruent, okay? So now we're ready to introduce the congruence axioms, the remaining three. So C4, C5, C6, due to Hilbert's axioms of Euclidean geometry. The first couple, the first two are actually gonna be very, very similar to what we saw before. So we're gonna take axiom C4 to be angle translation. It's gonna be similar to segment translation and this is how it's gonna go. Given an angle ABC and given a ray B prime, A prime, then there exists a unique ray B prime, C prime on each side of the line, A prime, B prime, such that the angle ABC is congruent to the angle A prime, B prime, C prime. In particular, on the ray BA and on the side containing the point C, that unique ray coincides with BC. So all right, there's a lot to unravel here. So we have some angle ABC. So imagine we have something like this. Draw in some arrows because these are rays. So we have some point A. B is the vertex of this thing. C is some other point. So the angle ABC is given here. And so then we have some ray B prime A. So we might get something like the following. There's some ray A prime, B prime, like so. So the ray emanates from B prime. Oh, excuse me, this is supposed to be the, nope, that was right, B prime, A prime. So then we're given the angle we're given a ray segment or angle translation means that we can copy this angle down onto this ray that's below. So there exists some ray, some point, I should say. Well, no, there's some ray B prime, C prime. So the ray emanates from B prime as well. There's some point C prime right here. And so then we're gonna get that this angle is congruent to that one. ABC is congruent to A prime, B prime, C prime, okay? And so just like segment translation, angle translation has this uniqueness statement that if you translate an angle onto itself, the congruence in that situation actually has to be equality, right? So if you take, if we take the angle ABC and you translate it onto the ray AB, then the ray that's guaranteed to give you the congruent angle must have been BC, like so. Now with, unlike a segment translation, angle translation actually guarantees the angle can be translated in two different places because after all, when you have the ray B prime A, this ray could, of course, be extended to a line and we know from order geometry that there are two sides of the line. There's this side and that side. So when it comes to angle translation, we can actually translate onto either side of the line. It doesn't really matter. So we get this angle A prime, B prime, C prime, or this angle A prime, B prime, C double prime, they're both gonna be congruent to the original angle ABC. So when you work with the angle translation, you do have to, of course, pick which side of the line you're doing and on each side, there'll be a unique ray that gives you a congruent angle. And of course, when you copy it back onto yourself, that has to be the exact same thing, the exact same rays. You get equality in that situation, all right? So angle translation sounds a lot like segment translation, but we have to change the appropriate parts so that it's compatible with angles as opposed to segments. Axiom C5, which we will call transitivity of angle congruence, it sounds a lot like transitivity of segment congruence. We get that if the angle ABC is congruent to the angle A prime, B prime, C prime, and the angle A prime, B prime, C prime is congruent to the angle A double prime, B double prime, C double prime, then we can conclude that the angle ABC is congruent to the angle A double prime, B double prime, and C double prime, like so. Now, if we were just to stop right here for one moment, I'm not gonna even introduce the third axiom, the sixth axiom, I should say, just quite yet. We will do that, of course, in a moment. But let me scroll up so we can see the angle translation. Angle translation is essentially the same axiom but for angles as opposed to segments. And in particular, we have this uniqueness statement, right? This ray is gonna be unique, there's only one of them. And of course, if you translate onto yourself, you get equality. So that's built into the cake there, we have transitivity here. Now, this uniqueness statement is very, very important because when we talked about segment congruence, we used the uniqueness statement of segment translation to argue and prove that the relation of segment congruence is a reflexive, a reflexive relation. Because angle translation likewise has that uniqueness statement, the analogous uniqueness statement, we can infer that by the exact same proof that angle congruence is likewise gonna be a reflexive relation. By axioms, we have that it's transitive. Is it symmetric? Think about this, when we introduced the segment congruence with its three axioms, we then proved as a proposition that segment congruence is actually a symmetric relationship. And how do we do that? Well, I'm not gonna go through all of the arguments here but basically argument came down to we used segment translation, then we used transitivity of segment congruence and then we used the uniqueness of segment translation. So we use segment translation again to argue that segment congruence is a symmetric relationship. And since you're reflexive, symmetric and transitive, this then proves that segment congruence is an equivalence relation. I am gonna leave it as an exercise to the viewer here to prove that angle congruence is likewise an equivalence relation because the uniqueness statement of angle translation will give us angle reflexivity. Transitivity, of course, falls from the axiom and then the proof of angle symmetry with regard to congruence will come about from using angle translation, right? You take one of the angles translated to the other, use transitivity there and then you use the uniqueness to get that you started off with the same angle. So without any further proof, I can actually say that angle congruence is an equivalence relation. And this is one of the powers of the axiomatic method because angle congruence and segment congruence essentially have the same axioms, the same proof applies and we get an equivalence relation on angle congruence, okay? The last of the congruence axioms, axiom C6, it's gonna be a little bit different. If you recall for segment congruence, our last axiom was segment addition allowed us to unite two segments together and guarantee statements of congruence. It would make sense to have an analogous statement for angle congruence here, like a so-called angle addition for which we will have angle addition as a theorem later on. We don't wanna take it as its axiom because if we left it alone, if we took angle addition as the axiom, then we have this problem. Yeah, we can develop segment congruence on one side. We can develop angle congruence on a different side but how do they mesh together? How do they interact with one another? After all, the whole point of segment addition was to see how segment congruence interacts with betweenness of points. We have to make a connection to this new undefined term of congruence with the old undefined terms of incidence and betweenness. So in particular, we need some way of connecting angle congruence to segment congruence. And so instead of taking angle addition as an axiom, we'll prove it as a theorem and instead we're gonna take as our axiom, the side angle, side axiom, which this is gonna be a triangle congruent statement. So imagine we have two triangles, ABC and A prime, B prime, C prime. So we have these two triangles here and I'm gonna scooch things up a little bit so we can try to draw a picture to give some illustration what's going on here. So we have two triangles. Here's one triangle and I'll label it. So this is A, B and C. And then we have some other triangle. Giving a little bit more space here. We have some other triangle, maybe something like this. And so we'll call this one A prime, this one B prime and this one C prime. So we have these two triangles, these six points here. We're gonna assume that the side of the segment AB is congruent to the segment A prime, B prime. We're gonna assume that the angle ABC is congruent to the angle A prime, B prime, C prime. And we're gonna assume that the segment BC is congruent to B prime, C prime. And this is where it gets the name side angle side, often denoted AS, excuse me, SAS for short, side angle side, because we have this side that's congruent, the angle's congruent and then the side here. And in particular, the angle is between the two sides that's very important. Because if we had like some angle side side or something like that, that gets a little bit more complicated. We'll talk about that some other time. But the side angle side axiom then says that if we have this congruent, side angle side congruence between the triangles, then in fact, the triangles are actually congruent to each other. And so remember our definition of congruence of triangles. If the triangles are congruent, that means all of their sides must be congruent, right? So the other side, we don't know anything about AC and A prime, C prime. If the triangles were congruent, then we can infer that those sides are congruent. Likewise, the angles, the angle C would have to be congruent to angle C prime and angle A would have to be congruent to A prime, right? So these red marks we see right here, if we know those are congruent, then we can assume the other parts are congruent as well. And that's the power of these triangle congruences. If we know some of the parts are congruent, then we can know the other parts are congruent. And you're gonna see us because the side angle side and its consequences, many of our geometric proofs and congruence geometry will be things like, oh, we're gonna prove that these two triangles are congruent to each other because of some side angle, side condition or something else. And then we'll take some unknown side or some unknown angle and then say, oh, these sides are congruent because corresponding parts of congruent triangles are congruent. We're gonna use that observation all the time. And so that's a very, very powerful. The side angle side axiom is very, very powerful. We will approve angle addition in the future, not in this video because side angle side is actually more powerful and in particular it connects together congruence of segments and congruence of angles. Because how are they related? Side segment congruence and angle congruence? Well, that happens from, of course, triangle congruence, which the side angle side axiom offers us.