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and the diagram at left I drew it up here above or I pasted it up here line segment ad intersects B e in line segment a B a B is parallel to de what is the length of side C E so we want to figure out the length of side we want to figure out the length of side C E now it might jump out at you at least it looks like it immediately that triangle the small triangle on the left is going to be similar to this larger triangle on the right but let's let's feel good about that let's prove that to ourselves and to help ourselves prove that we're gonna remind ourselves that a B is parallel to D E and actually to think about that even more I'm gonna I'm gonna make I'm gonna extend these segments a little bit so that you can really start to see that they are parallel lines and that we have a transversal or we have two transversals that intersect those parallel lines all right and I could keep if I wanted to extend this as a line I could keep going in both directions but the bottom line are not to be funny about it is that these two things are parallel that is parallel to that and that's going to help us establish that these two triangles are that these two triangles are similar well the first thing we know is look this angle is vertical is a vertical angle with this angle so they're going to be congruent this angle right over here if you view segment B E as a transversal of the two parallel lines that that right over there is going to be an alternate interior angle this is going to be an alternate interior angle to this angle right over here these are alternate interior alternate interior angles so they're going to be congruent if some of this sounds like Greek to you and you are not Greek I encourage you to watch the videos on Khan Academy of angles of transversals and parallel lines and by the same argument if we look at ad as a transversal this angle right over here let me do it in a different color this angle this angle right over here is an alternate interior angle with this one so they're going to be congruent but the whole point of everything I just did is to show that look the three angles here are congruent to the three angles here so these must be similar triangles and the reason why that's useful is because if the two triangles are similar then the ratio between corresponding sides is going to be the same so for example the ratio of side seee so we could say the ratio the ratio of seee segment of the length of segment C e to say to say this side right over here so side segment AED which has length 7.5 is going to be equal to the ratio of the corresponding side to see it looks e e is the side opposite this green angle so if you go on this similar triangle opposite to this green angle that's segment BC so it's going to be equal to the ratio of the corresponding side so 2.1 and the other triangle to the corresponding side of 7 of d of de in the other triangle and this side is opposite this blue angle this this one of the vertical angles right over here so the corresponding side over here is going to be side VA so this is gonna be over 2.5 so what would Cee be equal to well multiple ways to solve this but one way we could just multiply both sides by 7 point 5 multiply both sides by 7 point 5 now what's seven point five divided by two point five well 75/25 would be three so seven 7 point 5 divided by 2 point 5 is 3 so this is going to be 3 so length of segment C is two point one times three so the length of segment c e is going to be what is that six point three and if we look down here we see that that is indeed that is indeed one of the choices