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This will show you the basics of triangle similarity (less than 10 minute video)

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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.

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Intro to triangle similarity (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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This is a direct link to SAS chapter in Similarity Hickman 246 Book

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This book was created for the investigation of similarity in triangles and polygons

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This will introduce the three basic postulates AA, SAS, and SSS similarity postulates (less than 13 minutes)

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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.

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Triangle similarity postulates/criteria (video) | Khan Academy

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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.

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This is a direct link to AA chapter in Similarity Hickman 246 Book

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This book was created for the investigation of similarity in triangles and polygons

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Khan Academy SELF CHECK on AA similarity

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Khan Academy SELF CHECK on AA similarity

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This is practice from Khan on AA similarity

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Determine similar triangles: Angles (practice) | Khan Academy

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Given two triangles with some of their angle measures, determine whether the triangles are similar or not.

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This is a direct link to SSS chapter in Similarity Hickman 246 Book

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This book was created for the investigation of similarity in triangles and polygons

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This is practice for SSS similarity

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Determine similar triangles: SSS (practice) | Khan Academy

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Given two triangles with their side measures, determine whether the triangles are similar or not.

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What I want to do in this video is see if we can identify similar triangles here and prove to ourselves that they really are similar, using some of the postulates that we've set up. So over here, I have triangle BDC. It's inside of triangle AEC. They both share this angle right over there, so that gives us one angle. We need two to get to angle-angle, which gives us similarity. And we know that these two lines are parallel. We know if two lines are parallel and we have a transversal that corresponding angles are going to be congruent. So that angle is going to correspond to that angle right over there. And we're done. We have one angle in triangle AEC that is congruent to another angle in BDC, and then we have this angle that's obviously congruent to itself that's in both triangles. So both triangles have a pair of corresponding angles that are congruent, so they must be similar. So we can write, triangle ACE is going to be similar to triangle-- and we want to get the letters in the right order. So where the blue angle is here, the blue angle there is vertex B. Then we go to the wide angle, C, and then we go to the unlabeled angle right over there, BCD. So we did that first one. Now let's do this one right over here. This is kind of similar, but it looks, just superficially looking at it, that YZ is definitely not parallel to ST. So we won't be able to do this corresponding angle argument, especially because they didn't even label it as parallel. And so you don't want to look at things just by the way they look. You definitely want to say, what am I given, and what am I not given? If these weren't labeled parallel, we wouldn't be able to make the statement, even if they looked parallel. One thing we do have is that we have this angle right here that's common to the inner triangle and to the outer triangle, and they've given us a bunch of sides. So maybe we can use SAS for similarity, meaning if we can show the ratio of the sides on either side of this angle, if they have the same ratio from the smaller triangle to the larger triangle, then we can show similarity. So let's go, and we have to go on either side of this angle right over here. Let's look at the shorter side on either side of this angle. So the shorter side is two, and let's look at the shorter side on either side of the angle for the larger triangle. Well, then the shorter side is on the right-hand side, and that's going to be XT. So what we want to compare is the ratio between-- let me write it this way. We want to see, is XY over XT equal to the ratio of the longer side? Or if we're looking relative to this angle, the longer of the two, not necessarily the longest of the triangle, although it looks like that as well. Is that equal to the ratio of XZ over the longer of the two sides-- when you're looking at this angle right here, on either side of that angle, for the larger triangle-- over XS? And it's a little confusing, because we've kind of flipped which side, but I'm just thinking about the shorter side on either side of this angle in between, and then the longer side on either side of this angle. So these are the shorter sides for the smaller triangle and the larger triangle. These are the longer sides for the smaller triangle and the larger triangle. And we see XY. This is two. XT is 3 plus 1 is 4. XZ is 3, and XS is 6. So you have 2 over 4, which is 1/2, which is the same thing as 3/6. So the ratio between the shorter sides on either side of the angle and the longer sides on either side of the angle, for both triangles, the ratio is the same. So by SAS we know that the two triangles are congruent. But we have to be careful on how we state the triangles. We want to make sure we get the corresponding sides. And I'm running out of space here. Let me write it right above here. We can write that triangle XYZ is similar to triangle-- so we started up at X, which is the vertex at the angle, and we went to the shorter side first. So now we want to start at X and go to the shorter side on the large triangle. So you go to XTS. XYZ is similar to XTS. Now, let's look at this right over here. So in our larger triangle, we have a right angle here, but we really know nothing about what's going on with any of these smaller triangles in terms of their actual angles. Even though this looks like a right angle, we cannot assume it. And if we look at this smaller triangle right over here, it shares one side with the larger triangle, but that's not enough to do anything. And then this triangle over here also shares another side, but that also doesn't do anything. So we really can't make any statement here about any kind of similarity. So there's no similarity going on here. There are some shared angles. This guy-- they both share that angle, the larger triangle and the smaller triangle. So there could be a statement of similarity we could make if we knew that this definitely was a right angle. Then we could make some interesting statements about similarity, but right now, we can't really do anything as is. Let's try this one out, this pair right over here. So these are the first ones that we have actually separated out the triangles. So they've given us the three sides of both triangles. So let's just figure out if the ratios between corresponding sides are a constant. So let's start with the short side. So the short side here is 3. The shortest side here is 9 square roots of 3. So we want to see whether the ratio of 3 to 9 square roots of 3 is equal to the next longest side over here, is 3 square roots of 3 over the next longest side over here, which is 27. And then see if that's going to be equal to the ratio of the longest side. So the longest side here is 6, and then the longest side over here is 18 square roots of 3. So this is going to give us-- let's see, this is 3. Let me do this in a neutral color. So this becomes 1 over 3 square roots of 3. This becomes 1 over root 3 over 9, which seems like a different number, but we want to be careful here. And then this right over here-- if you divide the numerator and denominator by 6, this becomes a 1 and this becomes 3 square roots of 3. So 1 over 3 root 3 needs to be equal to square root of 3 over 9, which needs to be equal to 1 over 3 square roots of 3. At first they don't look equal, but we can actually rationalize this denominator right over here. We can show that 1 over 3 square roots of 3, if you multiply it by square root of 3 over square root of 3, this actually gives you in the numerator square root of 3 over square root of 3 times square root of 3 is 3, times 3 is 9. So these actually are all the same. This is actually saying, this is 1 over 3 root 3, which is the same thing as square root of 3 over 9, which is this right over here, which is the same thing as 1 over 3 root 3. So actually, these are similar triangles. So we can actually say it, and I'll make sure I get the order right. So let's start with E, which is between the blue and the magenta side. So that's between the blue and the magenta side. That is H, right over here. I'll do it like this. Triangle E, and then I'll go along the blue side, F. Actually, let me just write it this way. Triangle EFG, we know is similar to triangle-- So E is between the blue and the magenta side. Blue and magenta side-- that is H. And then we go along the blue side to F, go along the blue side to I, and then you went along the orange side to G, and then you go along the orange side to J. So triangle EFJ-- EFG is similar to triangle HIJ by side-side-side similarity. They're not congruent sides. They all have just the same ratio or the same scaling factor. Now let's do this last one, right over here. Let's see. We have an angle that's congruent to another angle right over there, and we have two sides. And so it might be tempting to use side-angle-side, because we have side-angle-side here. And even the ratios look kind of tempting, because 4 times 2 is 8. 5 times 2 is 10. But it's tricky here, because they aren't the same corresponding sides. In order to use side-angle-side, the two sides that have the same corresponding ratios, they have to be on either side of the angle. So in this case, they are on either side of the angle. In this case, the 4 is on one side of the angle, but the 5 is not. So because if this 5 was over here, then we could make an argument for similarity, but with this 5 not being on the other side of the angle-- it's not sandwiching the angle with the 4-- we can't use side-angle-side. And frankly, there's nothing that we can do over here. So we can't make some strong statement about similarity for this last one.

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Determining similar triangles (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.