 In the previous video, we had proven that the side angle side triangle congruence criterion, which we take as an axiom of congruence geometry, implies the angle side angle congruence criterion. And in fact, we implied that the two are logically equivalent to each other. You could take either one equipped with the five other axioms of congruence and develop the exact same congruence theory. In this video, we are going to prove the side side side triangle criterion, which says that if you have two triangles, A, B, C, and A prime, B prime, C prime, if their corresponding sides are all congruent to each other, that is, if A, B is congruent to A prime, B prime, A, C is congruent to A prime, C prime, and B, C is congruent to B prime, C prime, then in that situation, we have that the triangles A, B, C are congruent to A prime, B prime, C prime. And the basic proof is going to follow from the side angle side condition or axiom, but we have to do some other things first. In particular, we're gonna use something called triangle translation. Now, by our congruence axioms, we have a segment translation and we have an angle translation that we can uniquely translate segments and angles to other points in the plane. Well, when you combine together segment translation, angle translation with side angle side, you can actually translate entire triangles. And I'm gonna leave that as an exercise to the viewer. In fact, this was actually a homework exercise for my students. So hopefully they've already run across triangle translation by now. But if not, go back to the lecture on angle translation. You should be able to do the exercise from there. So this proof is gonna be a little bit different than what we saw with angle side angle. What we're gonna see, of course, is our triangles, right? So I'm gonna put one of my triangles up here for the sake of reference, like so. So this will be our triangle ABC, A, B, and C, like so. And then we're gonna have our other triangle. I'm gonna put it down here, something like so. It's a different triangle, of course, A prime, C prime, and B prime, those should be our correspondence. So by assumption, we have that AC is congruent to A prime, C prime, BC is congruent to B prime, C prime, and A prime, B prime is congruent to AB. So we have those congruent statements going on there. We don't know anything about the angle congruences whatsoever, not yet. If we can get one angle congruence, then we can make an argument about side angle side. And that's basically the direction we're gonna go into. So by triangle translation, there's gonna exist some point C double prime, such that, and this will be on the opposite side of the line A prime, B prime, that C prime is. So these points are on opposite sides here. We can translate the triangle ABC onto the segment A prime, B prime. And we can do that because the segment A prime, B prime is congruent to the segment AB. So that's the hypothesis we need for triangle congruence. So our triangle ABC by translation, by construction here, is gonna be congruent to the triangle A prime, B prime, C double prime. So we have that congruence right there. Then of course by corresponding parts of congruent triangles are congruent, of course. We have that since these two triangles are congruent to each other, and this triangle is congruent to certain things about this triangle. We have some congruences as well. So in particular, we have that the segment AC is gonna be congruent to the segment A prime, C double prime. But AC is already by assumption congruent to the triangle A prime, C prime like so. So we can say that the line segment A prime, C prime is congruent to the line segment A prime, C double prime. And by similar reason, we can also say that the line segment B prime, C prime is congruent to B prime, C double prime. Again, by transitivity of congruence like so. So I want you to note here that in this situation, these two triangles right here, they have congruence between their line segments. These two triangles are congruent. So if I can prove that these two triangles are congruent by transitivity of congruence, these two triangles are gonna be congruent. And therefore I'm gonna focus on this diagram right here. Now, before say anything else about this, I didn't wanna make mention that this diagram is one of our first examples in our lecture series of a quadrilateral. Where a quadrilateral, we might say something like the quadrilateral A prime, C double prime, B prime, C double prime like so. A quadrilateral is just going to be a union of triangles. So this is the union of the triangle A prime, C prime, B prime with the triangle A prime, C double prime, B prime like so. But it's not just some arbitrary union of triangles. They have, it has to be a union for which the two triangles share a common edge, which in this situation, that common edge would be A prime, B prime, the segment. And they don't have any interior points that overlap. So basically like we have the following situation you see on the screen like so. So in general, if you think of a quadrilateral, you might think of something like the following. A quadrilateral can be dissected into two different triangles. Now, if it's a convex quadrilateral that you see right here, there's going to be two different ways you could dissect it. But if you have for example, a convex, excuse me, a concave quadrilateral, then there might be only one way to dissect it into two triangles. But that's fine, we get this notion of a quadrilateral. What we see on the screen right now is an example of a so-called kite. Where a kite is going to be a quadrilateral where these two sides are congruent to each other and these two sides are congruent to each other, although not all four segments are necessarily congruent. So two consecutive sides are congruent and the other two consecutive sides are congruent to each other. This is called a kite and you can imagine why it gets such a name. Little kids like to fly kites like this every once in a while, you know? And if you're big and frank, you get struck by lightning when you do this. So be very, very careful. So we're going to play around with the congruence notion of a kite basically to prove the side, side, side triangle congruence. Now by construction, C and C double prime, you know, return to the proof of side, side, side here, they're on opposite sides of the line A prime B prime. And so as such, if we look at the line segment that connects the two together, there's going to be some point of intersection between the segment C prime C double prime with the line A prime B prime. And so we'll call that point of intersection D prime. We don't have a D in play here, but everything is called, everything's called prime over in this picture. So we always will keep with that pattern there. Notice of course that the point D prime is necessarily going to be between A prime and B prime. If that, well, I should say if that's the case, then we're going to have that. We don't actually know that yet. I guess I shouldn't get, I shouldn't put the cart in front of the horse there. We don't necessarily know that yet because we could have things like the following, right? What if our kite is actually more like a arrowhead for which this line segment might actually be exterior to the triangle. D prime might actually be this point right here. One should be very cautious with our diagrams because when I drew this kite, essentially I'm assuming that the trying other, the kite is going to be convex, but what if it's concave? That might change some things. So really we should emphasize if right here, if D prime is between A prime and B prime like the diagram seems to suggest, then that means that D prime is an interior point to the angle A prime, C prime, B prime like so, right? That would match up with the diagram. It would also mean that D prime is an interior point to the angle A prime, C double prime, B prime like so. Again, like so. And these are all consequences of the between cross lemma. That's if we have this between the statement. So in that case, in that case of course, we would have that the ray C prime D, which let me get used a different color here, the ray C prime D would be between the ray C prime A prime and the ray C prime B prime like so. We would also have that the ray, the ray C double prime D prime sits between C double prime A prime and C double prime B prime. Again, supporting the diagram we have on the board here. But of course, again, this is all contingent upon this hypothesis. We'll come back to that one in just a moment. This is an appropriate time to introduce the so-called a Sossley's Triangle theorem. Remember in a Sossley's Triangle is a triangle for which two of the sides are congruent to each other. So if we look at the triangle A prime C prime C double prime, which by these statements about betweenness, means that our diagram is exactly what we think it is. We have this triangle A prime C prime C double prime. This is an a Sossley's Triangle. And so by the a Sossley's Triangle theorem, we actually are gonna get that these angles are congruent to each other. I'm gonna leave it as an exercise to the viewer here to my students to prove the a Sossley's Triangle theorem. But basically the idea is use the triangle and you do a side angle side argument on itself, just switching the roles of C prime and C double prime. And you can get this triangle congruent. So we get these angles are congruent to each other. But likewise, the triangle B prime C prime C double prime is likewise an a Sossley's Triangle. In which case we then get by the a Sossley's Triangle theorem that angle D prime C prime B prime is congruent to D prime C double prime B prime like so. And so then if we use angle addition, we can add these angles back, we can add these angles together to get that angle a prime C prime B prime is congruent to angle a prime C double prime B prime. So then we have this side angle side situation so that the blue triangle right here is congruent to the side angle side over here. That is the triangle A prime C prime B prime. Thus proving that these two triangles, since those are congruent, those would be congruent as well. So that all follows from this if that we see of course on the screen right now. This if like so, where D prime is between A prime and B prime, we get the, we get a congruent set situation. So let's consider the other possibility, okay? What if, well, without the loss of generality we can assume that if it's not this case, then it actually must be the case that B prime is between A prime and D prime. So we get a picture that would look something like the following. Again, I'm gonna try to reschedge this on the screen real quick. We might get, and so relabeling, we have our point A prime right here, C prime's right here, C double prime is right here, B prime is right here, and then D prime is some point, actually D prime would be this point right here. Whoops, let me do that as a straight edge. Like so, and so D prime would be this point right here. We of course are still assuming a congruence in this situation like so. And so what if the picture looked more like this, okay? In that situation, we would then have that B is an interior angle to, it's interior points to the angle D prime, C prime, A prime. So if you look at this one right here, this angle, D prime, I guess I did the other side, sorry about that, D prime, C prime, A prime, but it also would be interior to the angle, D prime, C double prime, A prime like so. And so this then gives us that the ray, C prime, B prime, this ray right here is between C prime, A prime, and C prime, D prime like so. We also get that the ray C prime, C double prime, B prime sits between C prime, D prime, and C double prime, A prime like so. And so then we wanna play around with the Asosceles Triangle theorem one more time. So if you look at the Asosceles Triangle, basically I'm arguing that, oh, this is the diagram we have in that situation. Again, we have an Asosceles Triangle here, so these angles are congruent to each other. And then looking at this triangle, A prime, C prime, C double prime like so, we get that these angles are congruent to each other. And so then using angle subtraction this time, we then are gonna get that this angle right here, the angle B prime, C prime, A prime, is congruent to this angle right here, B prime, C double prime, A prime like so. So then we again still get side angle side like so. So the triangle A prime, C prime, B prime is congruent to the angle, the triangle A prime, C double prime, B prime like so. So it's a different congruence, but in the end, side angle side saves the day. We also have to use this Asosceles Triangle theorem in the, well, secretly, right? Not really secretly, I told you all about it, but using, we haven't proven, we haven't proven side angle side because I left it as an exercise. We also did improve triangle translation. Many of these exercises are important that the viewer works through them so they understand the potential gaps in these proofs.