Proof: Parallel lines divide triangle sides proportionally

- [Instructor] We're asked to prove that if a line is parallel to one side of a triangle, then it divides the other two sides proportionally. So pause this video and see if you can do that. And you might wanna leverage this diagram. All right, so let's work through this together. So we can start with this diagram. And what we know is that segment ED is parallel to segment CB. So we can write that down. Segment ED is parallel to segment CB. And so segment ED is what they're talking about. That is a line or a line segment that is parallel to one side of the triangle. So really given what we know, and what's already been written over here on this triangle, we need to prove another way of writing it, another way of saying it divides the other two sides proportionately, is that the ratio between the part of the original triangle side that is on one side of the dividing line to the length on the other side is going to be the same on both sides that it is intersecting. So another way to say that it divides the other two sides proportionately, if we look at this triangle over here, it would mean that the length of segment AE over the length of segment EC is going to be equal to the length of segment AD over the length of segment DB. This statement right over here, and what I underlined up here are equivalent given this triangle. So the way that we can try to do it is to establish a similarity between triangle AED and triangle ACB. So how do we do that? Well, because these two lines are parallel, we can view segment AC as a transversal intersecting two parallel lines. So that tells us that these two corresponding angles are going to be congruent. So we could say that angle one is congruent to angle three. And the reason why, is because they are corresponding, corresponding angles. I'm just trying to write a little bit of shorthand. This is short for corresponding angles. That's the rationale. And we also know that angle two is congruent to angle four for the same reason. So angle two is congruent to angle four. Once again, because they are corresponding angles. This time we have a different transversal, corresponding angles where a transversal intersects two parallel lines. And so now, if you look at triangle AED and triangle ACB, you see that they have two sets of corresponding angles that are congruent. And if you have two sets of corresponding angles, that means that all of the angles are congruent. And you actually see that over here, if you care about it, but two is enough, but you actually have a third because angle, I guess you call it BAC is common to both triangles. And so we can say that triangle AED is similar to triangle ACB, ACB by angle similarity. Similarity. And then given that these two are similar, then we can set up a proportion. That tells us that the ratio of the length of segment AE to this entire side to AC, is equal to the ratio of AD, the length of that segment to the length of the entire thing, to AB. Now this implies, and I'm just gonna start writing it to the right here to save space. This is the same thing as the ratio of AE over, AC is AE plus EC's length. So AE's length plus EC's length, and then this is going to be equal to the length of segment AD over segment AB, its length is the length of segment AD, AD plus segment DB, plus D B. Now, really what I need to do is figure out how do I algebraically manipulate it so I get what I have up here. Let me scroll down a little bit. So one way I could try to simplify this is to essentially cross multiply, that's equivalent to multiplying both sides by both of these denominators. And we've covered that in other videos. And so this is going to be equal to the length of segment AE times AD plus DB, those segment lengths, that's gotta be equal to length of AD times AE plus the length of segment EC. And I can distribute this over here. I have length of segment AE times length of segment AD plus length of segment AE times length of segment DB is equal to length of segment AD times length of segment AE plus length of segment AD times length of segment EC. And let's see, is there anything that I can simplify here? We'll have AE times AD on both sides. So let me just subtract AE times AD from both sides. And so then I'm just left with this is equal to that. So scroll down a little bit more and let me actually just rewrite this cleanly. So I have AE times DB is equal to AD times EC. These are all the segment lengths right over here. Now, if you divide both sides by EC, you're going to get an EC down here, and then this would cancel out. And then if you divide both sides by DB, this will cancel out and you'll get a DB right over here. So if you just algebraically manipulate what we just had over there, you get that the length of segment AE over the length of segment EC, length of segment AE over the length of segment EC is equal to the length of segment AD, AD over the length of segment DB, which is exactly what we wanted to prove. That this line right over here that is parallel to this side over here, divides the other two sides proportionally.