Introduction to triangle similarity
When we compare triangle ABC to triangle XYZ, it's pretty clear that they aren't congruent, that they have very different lengths of their sides. But there does seem to be something interesting about the relationship between these two triangles. One, all of their corresponding angles are the same. So the angle right here, angle BAC, is congruent to angle YXZ. Angle BCA is congruent to angle YZX, and angle ABC is congruent to angle XYZ. So all of their corresponding angles are the same. And we also see that the sides are just scaled-up versions of each other. So to go from the length of XZ to AC, we can multiply by 3. We multiplied by 3 there. To go from the length of XY to the length of AB, which is the corresponding side, we are multiplying by 3. We have to multiply by 3. And then to go from the length of YZ to the length of BC, we also multiplied by 3. So essentially, triangle ABC is just a scaled-up version of triangle XYZ. If they were the same scale, they would be the exact same triangles. But one is just a bigger, a blown-up version of the other one. Or this is a miniaturized version of that one over there. If you just multiply all the sides by 3, you get to this triangle. And so we can't call them congruent, but this does seem to be a bit of a special relationship. So we call this special relationship similarity. So we can write that triangle ABC is similar to triangle-- and we want to make sure we get the corresponding sides right-- ABC is going to be similar to XYZ. And so, based on what we just saw, there's actually kind of three ideas here. And they're all equivalent ways of thinking about similarity. One way to think about it is that one is a scaled-up version of the other. So scaled-up or -down version of the other. When we talked about congruency, they had to be exactly the same. You could rotate it, you could shift it, you could flip it. But when you do all of those things, they would have to essentially be identical. With similarity, you can rotate it, you can shift it, you can flip it. And you can also scale it up and down in order for something to be similar. So for example, let's say triangle CDE, if we know that triangle CDE is congruent to triangle FGH, then we definitely know that they are similar. They are scaled up by a factor of 1. Then we know, for a fact, that CDE is also similar to triangle FGH. But we can't say it the other way around. If triangle ABC is similar to XYZ, we can't say that it's necessarily congruent. And we see, for this particular example, they definitely are not congruent. So this is one way to think about similarity. The other way to think about similarity is that all of the corresponding angles will be equal. So if something is similar, then all of the corresponding angles are going to be congruent. I always have trouble spelling this. It is 2 Rs, 1 S. Corresponding angles are congruent. So if we say that triangle ABC is similar to triangle XYZ, that is equivalent to saying that angle ABC is congruent-- or we could say that their measures are equal-- to angle XYZ. That angle BAC is going to be congruent to angle YXZ. And then finally, angle ACB is going to be congruent to angle XZY. So if you have two triangles, all of their angles are the same, then you could say that they're similar. Or if you find two triangles and you're told that they are similar triangles, then you know that all of their corresponding angles are the same. And the last way to think about it is that the sides are all just scaled-up versions of each other. So the sides scaled by the same factor. In the example we did here, the scaling factor was 3. It doesn't have to be 3. It just has to be the same scaling factor for every side. If we scaled this side up by 3 and we only scaled this side up by 2, then we would not be dealing with a similar triangle. But if we scaled all of these sides up by 7, then that's still a similar, as long as you have all of them scaled up or scaled down by the exact same factor. So one way to think about it is-- I want to still visualize those triangles. Let me redraw them right over here a little bit simpler. Because I'm not talking in now in general terms, not even for that specific case. So if we say that this is A, B, and C, and this right over here is X, Y, and Z. I just redrew them so I can refer them when we write down here. If we're saying that these two things right over here are similar, that means that corresponding sides are scaled-up versions of each other. So we could say that the length of AB is equal to some scaling factor-- and this thing could be less than 1-- some scaling factor times the length of XY, the corresponding sides. And I know that AB corresponds to XY because of the order in which I wrote this similarity statement. So some scaling factor times XY. We know that the length of BC needs to be that same scaling factor times the length of YZ. And then we know the length of AC is going to be equal to that same scaling factor times XZ. So that's XZ, and this could be a scaling factor. So if ABC is larger than XYZ, then these k's will be larger than 1. If they're the exact same size, if they're essentially congruent triangles, then these k's will be 1. And if XYZ is bigger than ABC, then these [? scaling ?] factors will be less than 1. But another way to write these same statements-- notice, all I'm saying is corresponding sides are scaled-up versions of each other. This first statement right here, if you divide both sides by XY, you get AB over XY is equal to our scaling factor. And then the second statement right over here, if you divide both sides by YZ-- let me do it in that same color-- you get BC divided by YZ is equal to that scaling factor. And remember, in the example we just showed, that scaling factor was 3. But now we're saying in the more general terms, similarity, as long as you have the same scaling factor. And then finally, if you divide both sides here by the length between X and Z, or segment XZ's length, you get AC over XZ is equal to k, as well. Or another way to think about it is the ratio between corresponding sides. Notice, this is the ratio between AB and XY. The ratio between BC and YZ, the ratio between AC and XZ, that the ratio between corresponding sides all gives us the same constant. Or you could rewrite this as AB over XY is equal to BC over YZ is equal to AC over XZ, which would be equal to some scaling factor, which is equal to k. So if you have similar triangles-- let me draw an arrow right over here. Similar triangles means that they're scaled-up versions, and you can also flip and rotate and do all the stuff with congruency. And you can scale them up or down. Which means all of the corresponding angles are congruent, which also means that the ratio between corresponding sides is going to be the same constant for all the corresponding sides. Or the ratio between corresponding sides is constant.