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Congruent triangles & the SSS postulate/criterion

CCSS Math: HSG.CO.B.7

Video transcript

Let's talk a little bit about congruence. And one way to think about congruence, it's really a kind of equivalence for shape. So in algebra when something is equal to another thing, it means that their quantities are the same. But if we're now all of a sudden talking about shapes, and we say that those shapes are the same size and shape, then we say that they're congruent. And just to see a simple example here. I have this triangle right over there. And let's say I have this triangle right over here. And if you are able to shift this triangle and rotate this triangle and flip this triangle, you can make it look exactly like this triangle. As long as you're not changing the lengths of any of the sides or the angles here. But you can flip it, you can shift it, and rotate it. Let me write this. You can shift it, you can flip it, and you can rotate. If you can do those three procedures to make these the exact same triangle, to make them look exactly the same, then they are congruent. And if you say that a triangle is congruent-- let me label these. So let's call this triangle A, B, and C. And let's call this X, Y, and Z. So if we were to say, if we make the claim, that both of these triangles are congruent, so if we say triangle ABC is congruent-- and the way you specify it, it looks almost like an equal sign, but it's an equal sign with this little curly thing on top. Let me write a little bit neater. So we would write it like this. If we know that triangle ABC is congruent to triangle XYZ, that means that their corresponding sides have the same length, and their corresponding angles have the same measure. So if we make this assumption, or if someone tells us that this is true, then we know, for example, that AB is going to be equal to XY. The length of segment AB is going to be equal to the length of segment XY. And we could denote it like this. And I'm assuming that these are the corresponding sides. And you can see it, actually, by the way we've defined these triangles. A corresponds to X, B corresponds to Y, and then C corresponds to Z right over there. So side AB is going to have the same length as side XY. And you can sometimes, if you don't have the colors, you would denote it just like that. These two line segments have the same length. And you could actually say this, and you don't always see it written this way, you could also make the statement that line segment AB is congruent to line segment XY. But congruence of line segments really just means that their lengths are equivalent. So these two things mean the same thing. If one line segment is congruent to another line segment, that just means the measure of one line segment is equal to the measure of the other line segment. And so we can go through all the corresponding sides if these two characters are congruent. We also know the length of BC is going to be the length of YZ, assuming that those are the corresponding sides. And we could put these double hash marks right over here to show that these two lengths are the same. And then if we go to the third side, we also know that these are going to have the same length, or the line segments themselves are going to be congruent. So we also know that the length of AC is going to be equal to the length of XZ. Not only do we know that all the corresponding sides are going to have the same length, if someone tells us that a triangle is congruent, we also know that all the corresponding angles are going to have the same measure. So for example, we also know that this angle's measure is going to be the same as the corresponding angle's measure. The corresponding angle is right over here. It's between this orange side and this blue side. Or this orange side and this purple side, I should say. In between this orange side and this purple side. And so it also tells us that the measure of angle BAC is equal to the measure of angle YXZ. Let me write that angle symbol a little less like a-- We could also write that, as angle BAC is congruent to angle YXZ. And once again, like line segments, if one line segment is congruent to another line segment, it just means that their lengths are equal. And if one angle is congruent to another angle, it just means that their measures are equal. So we know that those two corresponding angles have the same measure, they're congruent. We also know that these two corresponding angles have the same measure, and I'll use a double arc to specify that this has the same measure as that. So we also know that the measure of angle ABC is equal to the measure of angle XYZ. And then finally, we know that this angle, if we know that these two characters are congruent, that this angle is going to have the same measure as this angle, as its corresponding angle. So we know that the measure of angle ACB is going to be equal to the measure of angle XZY. Now what we're going to concern ourselves a lot with, is how do we prove congruence? Because it's cool, because if you can prove congruence of two triangles, then all of a sudden you can make all of these assumptions. And what we're going to find out, and this is going to be, we're going to assume it for the sake of kind of an introductory geometry course that this is an axiom, or postulate, or just something that you assume. And let me actually just write this down. So an axiom, very fancy word. Postulate, also a very fancy word. It really just means things that we're going to assume are true. An axiom is sometimes, there's a little bit of distinction sometimes, where someone will say an axiom is something that's self-evident, or it seems like a universal truth that is definitely true. And we just take it for granted. You can't prove an axiom. A postulate kind of has that same role, but sometimes you could say, well, let's just assume this is true and see, if we assume that that's true, what we can derive from it. What we can prove if we assume it's true. But for the sake of a introductory geometry class, and really in most of mathematics today, these two words are used interchangeably. An axiom or postulate, just very fancy words for things that we take as a given. Things that we just will assume. We won't prove them. We will start with these assumptions, and then we're just going to build up from there. And one of the core ones that we'll see in geometry is the axiom, or the postulate, that if all the sides are congruent, or if the lengths of all the sides of the triangle are congruent, then we are dealing with congruent triangles. So it's sometimes called a side-side-side postulate or axiom. We're not going to prove it here. We're just going to take it as a given. So this literally stands for side-side-side. And what it tells us is, if we have two triangles-- and so let's say that's, and that's another triangle right over there-- and we know that corresponding sides are equal. So we know that this side right over here is equal in length to that side, we know that this side right over here is equal in length to that side right over there, and we know that this side over here is equal in length to this side over here. Then we know, and we're just going to take this as an assumption, and can build up off of this, we know that they are congruent. That these two triangles are congruent to each other. I didn't put any labels there, so it's kind of hard for me to refer to them. But these two are congruent triangles. What's powerful there is if we know all the corresponding sides are equal, then we know they're congruent, and then we can make all the other assumptions, which means that the corresponding angles are also equal. So that we know that that's going to be congruent to that, or have the same measure. That's going to have the same measure as that. And then that is going to have the same measure as that right over there. And to see why that is a reasonable axiom, or a reasonable assumption, or a reasonable postulate to start off with, let's start with one triangle. So let's say I have this triangle right over here. So it has that side, and then it has that side, and then it has that side right over here. And what I'm going to do is see, if I have another triangle that has the exact same side lengths, is there any way for me to construct a triangle with the same side lengths that is different, that can't be translated to this triangle through flipping, shifting, or rotating. So we assume this other triangle's going to have a side that's the same length as that one right over there. So I'll try to draw it like that. Roughly the same length. We know it's going to have a side that's that length. Let me put on this side just to make it look a little bit more interesting. So we know it's going to have a side like that. So I'm going to draw it roughly the same length, but I'm going to try to do it a different angle. Now we know it's going to have a side that looks like that. And so let me, I'll put it right over here. It's about that length right over there. And so clearly, this isn't a triangle. In order to make it a triangle, I'll have to connect this point to that point right over there. And really there's only two ways to do it. I can rotate it around that little hinge right over there. If I connect them over here, then I'm going to get a triangle that looks like this, which is really just a flipped-- am I visualizing it right? Yeah, just a flipped version. You can rotate it a little bit back this way. But then you'd have the magenta on this side and the yellow on this side. And you could flip it vertically, and then it would look exactly like this. Our other option to make these two points connect, is to rotate them out this way. And then the yellow side is going to be here, and then the magenta side is going to be right over here. That's not magenta. The magenta side is going to be just like that. And if we do that, then we actually just have to rotate it around to get that exact triangle. So this isn't a proof, but actually we're going to start assuming that this is an axiom. But hopefully you see that it's a pretty reasonable starting point, that if all of the sides, all the corresponding sides, of two different triangles are equal, then we know that they are congruent. We're just going to assume, it is an axiom that we're going to build off of, that they're congruent. And then we also know that all the corresponding angles are going to be equivalent.