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# समांतर रेखाएँ और सांगत कोणों की उपपत्ति

वीडियो की प्रतिलिपि

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.