 This video goes through a couple of parallel lines and proportional parts theorems that you'll use to solve some problems. Let's take a look at the first theorem together. So the first theorem states, if a line is parallel to one side of a triangle and it intersects the other two sides in two distinct points, then that line separates the sides into proportional segments. So in other words, this line separates this length into proportional parts that follow the same proportion as that length. So let's get a couple of proportion examples. One possible proportion is to say B divided by A equals C divided by D. Now the reciprocal is also a true proportion. So this theorem is very particular in saying that it separates these sides into proportional parts. In other words, it separates the two sides that are cut. So this length and that length are cut into proportional parts. It does not say anything about the other lengths. I'll color them green. So this theorem does not give us any information about E and F. Instead, we have to think about this shape as two overlapping triangles if we're to deal with E and F. Let me give you an example. So we have B divided by B plus A. In other words, B divided by B plus A, this whole side, is equal to E divided by F. Again, E is a side length of this smaller triangle and F is a side length of the larger triangle. One other possibility, instead of B over B plus A, we could use C and C plus D. Another theorem we have states if three or more lines intersect two transversals, then those lines are cut proportionally. In other words, B over A is equal to C over D. And there are some alternative ways we can write that as well. Here is another example. We can write the reciprocal. A over B is equal to D over C. And there are a couple other ways we can consider this. We could write A plus B. In other words, the entire length of that transversal between the two parallels. A over B over A is equal to C plus D over D. So that's a possibility. And again, we can also write its reciprocal, just like we wrote the reciprocal here. And then here's another way. We can talk about A over B, sorry, A plus B, this entire length divided by B is equal to C over D divided by C. So we have one last theorem, and this theorem is actually more of a definition. The mid-segment of a triangle. The mid-segment of a triangle connects the midpoint of one side to the midpoint of another side. In other words, this point is the midpoint of this side. And likewise, this point is the midpoint of this side. So that the mid-segment of a triangle, it's three characteristics. It's parallel to the third side. So we see these parallel markings. The length is half the length of the third side. In other words, if this is 16, this must be 8 or half as long. And then the angles that are created are going to be congruent to the third side. So this angle up in here, let's call that, I don't know, angle one is congruent to angle two because they're corresponding angles. And likewise, these angles are congruent because they're corresponding based on parallel lines. So that is mid-segment of a triangle.