Video Transcript: Solving Similar Triangles

Solving Similar Triangles 

In this first problem over here, we're asked to find out the length of this segment, segment CE.  And we have these two parallel lines. AB is parallel to DE. And then, we have these two  essentially transversals that form these two triangles. So let's see what we can do here. So the  first thing that might jump out at you is that this angle and this angle are vertical angles. So  they are going to be congruent. The other thing that might jump out at you is that angle CDE is  an alternate interior angle with CBA. So we have this transversal right over here. And these are  alternate interior angles, and they are going to be congruent. Or you could say that, if you  continue this transversal, you would have a corresponding angle with CDE right up here and  that this one's just vertical. Either way, this angle and this angle are going to be congruent. So  we've established that we have two triangles and two of the corresponding angles are the  same. And that by itself is enough to establish similarity. We actually could show that this angle and this angle are also congruent by alternate interior angles, but we don't have to. So we  already know that they are similar. And actually, we could just say it. Just by alternate interior  angles, these are also going to be congruent. But we already know enough to say that they are  similar, even before doing that. So we already know that triangle-- I'll color-code it so that we  have the same corresponding vertices. And that's really important-- to know what angles and  what sides correspond to what side so that you don't mess up your, I guess, your ratios or so  that you do know what's corresponding to what. So we know triangle ABC is similar to  triangle-- so this vertex A corresponds to vertex E over here. It's similar to vertex E. And then,  vertex B right over here corresponds to vertex D. EDC. Now, what does that do for us? Well,  that tells us that the ratio of corresponding sides are going to be the same. They're going to be  some constant value. So we have corresponding side. So the ratio, for example, the  corresponding side for BC is going to be DC. We can see it in just the way that we've written  down the similarity. If this is true, then BC is the corresponding side to DC. So we know that the length of BC over DC right over here is going to be equal to the length of-- well, we want to  figure out what CE is. That's what we care about. And I'm using BC and DC because we know  those values. So BC over DC is going to be equal to-- what's the corresponding side to CE? The  corresponding side over here is CA. It's going to be equal to CA over CE. This is last and the first. Last and the first. CA over CE. And we know what BC is. BC right over here is 5. We know what  DC is. It is 3. We know what CA or AC is right over here. CA is 4. And now, we can just solve for  CE. Well, there's multiple ways that you could think about this. You could cross-multiply, which  is really just multiplying both sides by both denominators. So you get 5 times the length of CE.  5 times the length of CE is equal to 3 times 4, which is just going to be equal to 12. And then we  get CE is equal to 12 over 5, which is the same thing as 2 and 2/5, or 2.4. So this is going to be 2  and 2/5. And we're done. We were able to use similarity to figure out this side just knowing that  the ratio between the corresponding sides are going to be the same. Now, let's do this problem right over here. Let's do this one. Let me draw a little line here to show that this is a different  problem now. This is a different problem. So in this problem, we need to figure out what DE is.  And we, once again, have these two parallel lines like this. And so we know corresponding  angles are congruent. So we know that angle is going to be congruent to that angle because  you could view this as a transversal. We also know that this angle right over here is going to be  congruent to that angle right over there. Once again, corresponding angles for transversal. And

also, in both triangles-- so I'm looking at triangle CBD and triangle CAE-- they both share this  angle up here. Once again, we could have stopped at two angles, but we've actually shown that all three angles of these two triangles, all three of the corresponding angles, are congruent to  each other. And once again, this is an important thing to do, is to make sure that you write it in  the right order when you write your similarity. We now know that triangle CBD is similar-- not  congruent-- it is similar to triangle CAE, which means that the ratio of corresponding sides are  going to be constant. So we know, for example, that the ratio between CB to CA-- so let's write  this down. We know that the ratio of CB over CA is going to be equal to the ratio of CD over CE.  And we know what CB is. CB over here is 5. We know what CA is. And we have to be careful  here. It's not 3. CA, this entire side is going to be 5 plus 3. So this is going to be 8. And we know  what CD is. CD is going to be 4. And so once again, we can cross-multiply. We have 5CE. 5 times  CE is equal to 8 times 4. 8 times 4 is 32. And so CE is equal to 32 over 5. Or this is another way to think about that, 6 and 2/5. Now, we're not done because they didn't ask for what CE is. They're asking for just this part right over here. They're asking for DE. So we know that this entire  length-- CE right over here-- this is 6 and 2/5. And so DE right over here-- what we actually have to figure out-- it's going to be this entire length, 6 and 2/5, minus 4, minus CD right over here.  So it's going to be 2 and 2/5. 6 and 2/5 minus 4 and 2/5 is 2 and 2/5. So we're done. DE is 2 and  2/5. 

Solving similar triangles: same side plays different roles 

In this problem, we're asked to figure out the length of BC. We have a bunch of triangles here,  and some lengths of sides, and a couple of right angles. And so maybe we can establish  similarity between some of the triangles. There's actually three different triangles that I can see here. This triangle, this triangle, and this larger triangle. If we can establish some similarity  here, maybe we can use ratios between sides somehow to figure out what BC is. So when you  look at it, you have a right angle right over here. So in triangle BDC, you have one right angle.  In triangle ABC, you have another right angle. If we can show that they have another  corresponding set of angles are congruent to each other, then we can show that they're similar.  And actually, both of those triangles, both BDC and ABC, both share this angle right over here.  So if they share that angle, then they definitely share two angles. So they both share that angle right over there. Let me do that in a different color just to make it different than those right  angles. They both share that angle there. And so we know that two triangles that have at least  two congruent angles, they're going to be similar triangles. So we know that triangle ABC-- We  went from the unlabeled angle, to the yellow right angle, to the orange angle. So let me write it this way. ABC. And we want to do this very carefully here because the same points, or the same vertices, might not play the same role in both triangles. So we want to make sure we're getting  the similarity right. White vertex to the 90 degree angle vertex to the orange vertex. That is  going to be similar to triangle-- so which is the one that is neither a right angle-- so we're  looking at the smaller triangle right over here. Which is the one that is neither a right angle or  the orange angle? Well it's going to be vertex B. Vertex B had the right angle when you think  about the larger triangle. But we haven't thought about just that little angle right over there. So we start at vertex B, then we're going to go to the right angle. The right angle is vertex D. And  then we go to vertex C, which is in orange. So we have shown that they are similar. And now 

that we know that they are similar, we can attempt to take ratios between the sides. And so  let's think about it. We know what the length of AC is. AC is going to be equal to 8. 6 plus 2. So  we know that AC-- what's the corresponding side on this triangle right over here? So you could  literally look at the letters. A and C is going to correspond to BC. The first and the third, first and the third. AC is going to correspond to BC. And so this is interesting because we're already  involving BC. And so what is it going to correspond to? And then if we look at BC on the larger  triangle, BC is going to correspond to what on the smaller triangle? It's going to correspond to  DC. And it's good because we know what AC, is and we know it DC is. And so we can solve for  BC. So I want to take one more step to show you what we just did here, because BC is playing  two different roles. On this first statement right over here, we're thinking of BC. BC on our  smaller triangle corresponds to AC on our larger triangle. And then in the second statement, BC on our larger triangle corresponds to DC on our smaller triangle. So in both of these cases. So  these are larger triangles and then this is from the smaller triangle right over here.  Corresponding sides. And this is a cool problem because BC plays two different roles in both  triangles. But now we have enough information to solve for BC. We know that AC is equal to 8.  6 plus 2 is 8. And we know the DC is equal to 2. That's given. And now we can cross multiply. 8  times 2 is 16 is equal to BC times BC-- is equal to BC squared. And so BC is going to be equal to  the principal root of 16, which is 4. BC is equal to 4. And we're done. And the hardest part about this problem is just realizing that BC plays two different roles and just keeping your head  straight on those two different roles. And just to make it clear, let me actually draw these two  triangles separately. So if I drew ABC separately, it would look like this. So this is my triangle,  ABC. And then this is a right angle. This is our orange angle. We know the length of this side  right over here is 8. And we know that the length of this side, which we figured out through this problem is 4. Then if we wanted to draw BDC, we would draw it like this. So BDC looks like this.  So this is BDC. That's a little bit easier to visualize because we've already-- This is our right  angle. This is our orange angle. And this is 4, and this right over here is 2. And I did it this way to  show you that you have to flip this triangle over and rotate it just to have a similar orientation.  And then it might make it look a little bit clearer. So if you found this part confusing, I  encourage you to try to flip and rotate BDC in such a way that it seems to look a lot like ABC.  And then this ratio should hopefully make a lot more sense.