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# Solving similar triangles

CCSS.Math:

## Video transcript

in this first problem over here we're asked to find out the length of this segment segment seee and we have these two parallel lines a B 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 saying 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 they have to two of the corresponding angles are the same and that by itself is enough to establish similarity you can actually 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 actually we could just say it just all by alternate interior angles these are also going to be congruent but we've already we already know enough to say that they are similar even before doing that so we already know that triangle I'll try to write it color-coded 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 a B triangle a BC ABC is similar is similar to triangle so a this vertex a corresponds to vertex E over here is similar to vertex E and then vertex B right over here corresponds to vertex D E d e d c e d c now what does that do for us well that tells us that the ratio of corresponding sides are going to have the same they're 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 the corresponding side for BC is going to be DC we can see it 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 bc bc the length of bc over DC right over here over DC is going to be equal to is going to be equal to the length of well we want to figure out what c e is that's what we care about and i'm using b c and d c because we know those values so bc over DC is going to be equal to is going to be equal to what's the corresponding side to see e c e the corresponding side over here C a is going to be equal to C a over C II see a over C e corresponding sides this is the last and the first last and the first C a over C E and we know what B C is B C right over here is 5 we know what DC is it is 3 we know what C a or AC is right over here C a is 4 and now we can just solve for C e so we can well there's multiple ways that you could think about this you could cross multiply and you get which is really just multiplying both sides by both the denominators so you get 5 times the length of C e 5 times the length of C is equal to 3 times 4 which is just going to be equal to 12 and then we get C e c e is equal to 12 over 5 is equal to 12 over 5 which is the same thing as 2 and 2/5 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 so that this is a different problem now this is a different problem so this problem we need to figure out what de is and we all once again have these two we 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 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 C ae they both share this angle up here they both share this angle up here so we've actually shown 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 so we now know we now know once again this is an important thing to do is to make sure that you get is that you write it in the right order when you write your similarity we now know that triangle c c b c b d c BD is similar is similar not congruent it is similar to triangle C see a ç å è ç ä é c AE which means that the ratio of corresponding sides are going to be constant so we know for example that the ratio between c b is going to be the same there's a ratio for of C B to C A let's write this down we know that the ratio of C B over C a is going to be equal to the ratio of CD C D over C e C D over C E and we know what CB is CB over here is 5 we know what C a is and we have to be careful here it's not 3c a this entire side is going to be 5 plus 3 so this is going to be 8 and we know what C D is C D is going to be 4 and so once again we can cross multiply we have 5 C five times see E is equal to 8 times 8 times 4 8 times 4 is 32 32 and so see E is equal to C e is equal to 32 32 over 5 or this is another way to think about that 6 and 2/5 6 and 2/5 now we're not done because they didn't ask for what C II is they're asking for just this part right over here they're asking for de so we know that this entire length C e 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 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