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 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 if L is parallel to M then then X is equal to Y what I want to do with this video is prove it the other way around I want to prove 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 go 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 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 I'm going to assume that X is equal to Y and and L is not parallel 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 em like this they're going to intersect by definition if something 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 we could also call the measure of that 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 zero length and so this line right over here is not going to be of zero length of this line segment between points between points a and b i guess we could say that a be the length of that line segment is greater than zero I think that's a fair assumption in either case a B is going to be greater than zero so when we assume that these two things are not parallel we form ourselves a nice little triangle here where a B 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 go 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 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 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 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 and but the L is not parallel to M we get this weird situation where we form this triangle and the angle at the intersection of those of those two lines that are definitely not parallel all of a sudden becomes 0 degrees 0 degrees but that's completely nonsensical if this was zero degrees that means that this triangle wouldn't open up at all which means that the length of a B would have to be 0 would have to be essentially you could call it maybe like a degenerate triangle it wouldn't even be a triangle it would be aligned 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 this line segment a B would have to be equal to 0 it would wouldn't be there or another contradiction that you could come up with would be that would be that this this 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 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 lead you you 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 not for L and M to be two different lines and for them not to be parallel and so we have proven our statements 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