證明 - 同位角相等意謂著兩線平行 (英)
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證明 - 同位角相等意謂著兩線平行 (英): 利用反證法證明同位角相等，兩線就平行
- 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 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 are parallel then the corresponding angles are equal and i want to show that if the corresponding
- angles are equal, then the lines are definitely parallel.
- 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 this isn't true, I'm going to assume it's not true
- 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 reality
- that would create, so if l and m are not parallel and they are different lines
- and they are going to intersect at some point
- this is line l, let me draw m like this, they are going to intersect.
- By definition if two lines are not parallel, they are going to intersect each other
- and that is going to be m and this thing that was a transversal I'll just draw it over here
- and then this is x, this is y, we're assuming that y is equal to x so we could also call the measure
- of that angle x. So given all this reality and we're assuming that in either case, that this is some
- distance that this line is not of zero length so this line right over here is not going to be of zero length
- or this line segment between points a and b I guess we could say AB the length of that line
- segment is greater than zero I think that's a fair assumption in either case
- AB 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
- where AB is one of the sides and the other two sides I guess we can call this point of intersection