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.