Video Transcript: Intro to Similar Triangles

Intro 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. 

Triangle similarity postulates/criteria 

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. 

Determining similar triangles 

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.