Introduction to triangle similarity
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Let's say we have triangle ABC. It looks something like this. I want to think about the minimum amount of information. I want to come up with a couple of postulates that we can use to determine whether another triangle is similar to triangle ABC. So we already know that if all three of the corresponding angles are congruent to the corresponding angles on ABC, then we know that we're dealing with congruent triangles. So for example, if this is 30 degrees, this angle is 90 degrees, and this angle right over here is 60 degrees. And we have another triangle that looks like this, it's clearly a smaller triangle, but it's corresponding angles. So this is 30 degrees. This is 90 degrees, and this is 60 degrees, we know that XYZ in this case, is going to be similar to ABC. So we would know from this because corresponding angles are congruent, we would know that triangle ABC is similar to triangle XYZ. And you've got to get the order right to make sure that you have the right corresponding angles. Y corresponds to the 90-degree angle. X corresponds to the 30-degree angle. A corresponds to the 30-degree angle. So A and X are the first two things. B and Y, which are the 90 degrees, are the second two, and then Z is the last one. So that's what we know already, if you have three angles. But do you need three angles? If we only knew two of the angles, would that be enough? Well, sure because if you know two angles for a triangle, you know the third. So for example, if I have another triangle that looks like this-- let me draw it like this-- and if I told you that only two of the corresponding angles are congruent. So maybe this angle right here is congruent to this angle, and that angle right there is congruent to that angle. Is that enough to say that these two triangles are similar? Well, sure. Because in a triangle, if you know two of the angles, then you know what the last angle has to be. If you know that this is 30 and you know that that is 90, then you know that this angle has to be 60 degrees. Whatever these two angles are, subtract them from 180, and that's going to be this angle. So in general, in order to show similarity, you don't have to show three corresponding angles are congruent, you really just have to show two. So this will be the first of our similarity postulates. We call it angle-angle. If you could show that two corresponding angles are congruent, then we're dealing with similar triangles. So for example, just to put some numbers here, if this was 30 degrees, and we know that on this triangle, this is 90 degrees right over here, we know that this triangle right over here is similar to that one there. And you can really just go to the third angle in this pretty straightforward way. You say this third angle is 60 degrees, so all three angles are the same. That's one of our constraints for similarity. Now, the other thing we know about similarity is that the ratio between all of the sides are going to be the same. So for example, if we have another triangle right over here-- let me draw another triangle-- I'll call this triangle X, Y, and Z. And let's say that we know that the ratio between AB and XY, we know that AB over XY-- so the ratio between this side and this side-- notice we're not saying that they're congruent. We're looking at their ratio now. We're saying AB over XY, let's say that that is equal to BC over YZ. That is equal to BC over YZ. And that is equal to AC over XZ. So once again, this is one of the ways that we say, hey, this means similarity. So if you have all three corresponding sides, the ratio between all three corresponding sides are the same, then we know we are dealing with similar triangles. So this is what we call side-side-side similarity. And you don't want to get these confused with side-side-side congruence. So these are all of our similarity postulates or axioms or things that we're going to assume and then we're going to build off of them to solve problems and prove other things. Side-side-side, when we're talking about congruence, means that the corresponding sides are congruent. Side-side-side for similarity, we're saying that the ratio between corresponding sides are going to be the same. So for example, let's say this right over here is 10. No. Let me think of a bigger number. Let's say this is 60, this right over here is 30, and this right over here is 30 square roots of 3, and I just made those numbers because we will soon learn what typical ratios are of the sides of 30-60-90 triangles. And let's say this one over here is 6, 3, and 3 square roots of 3. Notice AB over XY 30 square roots of 3 over 3 square roots of 3, this will be 10. What is BC over XY? 30 divided by 3 is 10. And what is 60 divided by 6 or AC over XZ? Well, that's going to be 10. So in general, to go from the corresponding side here to the corresponding side there, we always multiply by 10 on every side. So we're not saying they're congruent or we're not saying the sides are the same for this side-side-side for similarity. We're saying that we're really just scaling them up by the same amount, or another way to think about it, the ratio between corresponding sides are the same. Now, what about if we had-- let's start another triangle right over here. Let me draw it like this. Actually, I want to leave this here so we can have our list. So let's draw another triangle ABC. So this is A, B, and C. And let's say that we know that this side, when we go to another triangle, we know that XY is AB multiplied by some constant. So I can write it over here. XY is equal to some constant times AB. Actually, let me make XY bigger, so actually, it doesn't have to be. That constant could be less than 1 in which case it would be a smaller value. But let me just do it that way. So let me just make XY look a little bit bigger. So let's say that this is X and that is Y. So let's say that we know that XY over AB is equal to some constant. Or if you multiply both sides by AB, you would get XY is some scaled up version of AB. So maybe AB is 5, XY is 10, then our constant would be 2. We scaled it up by a factor of 2. And let's say we also know that angle ABC is congruent to angle XYZ. I'll add another point over here. So let me draw another side right over here. So this is Z. So let's say we also know that angle ABC is congruent to XYZ, and let's say we know that the ratio between BC and YZ is also this constant. The ratio between BC and YZ is also equal to the same constant. So an example where this 5 and 10, maybe this is 3 and 6. The constant we're kind of doubling the length of the side. So is this triangle XYZ going to be similar? Well, if you think about it, if XY is the same multiple of AB as YZ is a multiple of BC, and the angle in between is congruent, there's only one triangle we can set up over here. We're only constrained to one triangle right over here, and so we're completely constraining the length of this side, and the length of this side is going to have to be that same scale as that over there. And so we call that side-angle-side similarity. So once again, we saw SSS and SAS in our congruence postulates, but we're saying something very different here. We're saying that in SAS, if the ratio between corresponding sides of the true triangle are the same, so AB and XY of one corresponding side and then another corresponding side, so that's that second side, so that's between BC and YZ, and the angle between them are congruent, then we're saying it's similar. For SAS for congruency, we said that the sides actually had to be congruent. Here we're saying that the ratio between the corresponding sides just has to be the same. So for example SAS, just to apply it, if I have-- let me just show some examples here. So let's say I have a triangle here that is 3, 2, 4, and let's say we have another triangle here that has length 9, 6, and we also know that the angle in between are congruent so that that angle is equal to that angle. What SAS in the similarity world tells you is that these triangles are definitely going to be similar triangles, that we're actually constraining because there's actually only one triangle we can draw a right over here. It's the triangle where all the sides are going to have to be scaled up by the same amount. So there's only one long side right here that we could actually draw, and that's going to have to be scaled up by 3 as well. This is the only possible triangle. If you constrain this side you're saying, look, this is 3 times that side, this is 3 three times that side, and the angle between them is congruent, there's only one triangle we could make. And we know there is a similar triangle there where everything is scaled up by a factor of 3, so that one triangle we could draw has to be that one similar triangle. So this is what we're talking about SAS. We're not saying that this side is congruent to that side or that side is congruent to that side, we're saying that they're scaled up by the same factor. If we had another triangle that looked like this, so maybe this is 9, this is 4, and the angle between them were congruent, you couldn't say that they're similar because this side is scaled up by a factor of 3. This side is only scaled up by a factor of 2. So this one right over there you could not say that it is necessarily similar. And likewise if you had a triangle that had length 9 here and length 6 there, but you did not know that these two angles are the same, once again, you're not constraining this enough, and you would not know that those two triangles are necessarily similar because you don't know that middle angle is the same. Now, you might be saying, well there was a few other postulates that we had. We had AAS when we dealt with congruency, but if you think about it, we've already shown that two angles by themselves are enough to show similarity. So why worry about an angle, an angle, and a side or the ratio between a side? So why even worry about that? And we also had angle-side-angle in congruence, but once again, we already know the two angles are enough, so we don't need to throw in this extra side, so we don't even need this right over here. So these are going to be our similarity postulates, and I want to remind you, side-side-side, this is different than the side-side-side for congruence. We're talking about the ratio between corresponding sides. We're not saying that they're actually congruent. And here, side-angle-side, it's different than the side-angle-side for congruence. It's this kind of related, but here we're talking about the ratio between the sides, not the actual measures.