Parallel lines & corresponding angles proof

We know that if we have two lines that are parallel-- so let me draw those two parallel lines, l and m. So that's line l and line m. We know that if they are parallel, then if we were to draw a transversal that intersects both of them, that the corresponding angles are equal. So this is x, and this is y So we know that if l is parallel to m, then x is equal to y. What I want to do in this video is prove it the other way around. I want to prove-- So this is what we know. We know this. What I want to do is prove if x is equal to y, then l is parallel to m. So that we can go either way. If they're parallel, then the corresponding angles are equal. And I want to show if the corresponding angles are equal, then the lines are definitely parallel. And what I'm going to do is prove it by contradiction. So let's put this aside right here. This is our goal. I'm going to assume that this isn't true. I'm going to assume that it's not true. So I'm going to assume that x is equal to y and l is not parallel to m. So let's think about what type of a reality that would create. So if l and m are not parallel, and they're different lines, then they're going to intersect at some point. So let me draw l like this. This is line l. Let me draw m like this. They're going to intersect. By definition, if two lines are not parallel, they're going to intersect each other. And that is going to be m. And then this thing that was a transversal, I'll just draw it over here. So I'll just draw it over here. And then this is x. This is y. And we're assuming that y is equal to x. So we could also call the measure of this angle x. So given all of this reality, and we're assuming in either case that this is some distance, that this line is not of 0 length. And so this line right over here is not going to be of 0 length. Or this line segment between points A and B. I guess we could say that AB, the length of that line segment is greater than 0. I think that's a fair assumption in either case. AB is going to be greater than 0. So when we assume that these two things are not parallel, we form ourselves a nice little triangle here, where AB is one of the sides, and the other two sides are-- I guess we could label this point of intersection C. The other two sides are line segment BC and line segment AC. And we know a lot about finding the angles of triangles. So let's just see what happens when we just apply what we already know. Well first of all, if this angle up here is x, we know that it is supplementary to this angle right over here. So this angle over here is going to have measure 180 minus x. And then we know that this angle, this angle and this last angle-- let's call it angle z-- we know that the sum of those interior angles of a triangle are going to be equal to 180 degrees. So we know that x plus 180 minus x plus 180 minus x plus z is going to be equal to 180 degrees. Now these x's cancel out. We can subtract 180 degrees from both sides. And we are left with z is equal to 0. So if we assume that x is equal to y but that l is not parallel to m, we get this weird situation where we formed this triangle, and the angle at the intersection of those two lines that are definitely not parallel all of a sudden becomes 0 degrees. But that's completely nonsensical. If this was 0 degrees, that means that this triangle wouldn't open up at all, which means that the length of AB would have to be 0. Essentially, you could call it maybe like a degenerate triangle. It wouldn't even be a triangle. It would be a line. These two lines would have to be the same line. They wouldn't even form a triangle. And so this leads us to a contradiction. The contradiction is that this line segment AB would have to be equal to 0. It kind of wouldn't be there. Or another contradiction that you could come up with would be that these two lines would have to be the same line because there's no kind of opening between them. So either way, this leads to a contradiction. And since it leads to that contradiction, since if you assume x equals y and l is not equal to m, you get to something that makes absolutely no sense. You contradict your initial assumptions. Then it essentially proves that if x is equal to y, then l is parallel to m. Because we've shown that if x is equal to y, there's no way for l and m to be two different lines and for them not to be parallel. And so we have proven our statement. So now we go in both ways. If lines are parallel, corresponding angles are equal. If corresponding angles are equal, then the lines are parallel.