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# Angles formed between transversals and parallel lines

Angles of parallel lines. Created by Sal Khan.

Video transcript

In this video we're going to
think a little bit about parallel lines, and other lines
that intersect the parallel lines, and we call
those transversals. So first let's think about
what a parallel or what parallel lines are. So one definition we could use,
and I think that'll work well for the purposes of this video,
are they're two lines that sit in the same plane. And when I talk about a plane,
I'm talking about a, you can imagine a flat two-dimensional
surface like this screen -- this screen is a plane. So two lines that sit in a
plane that never intersect. So this line -- I'll try my
best to draw it -- and imagine the line just keeps going in
that direction and that direction -- let me do another
one in a different color -- and this line right
here are parallel. They will never intersect. If you assume that I drew it
straight enough and that they're going in the exact
same direction, they will never intersect. And so if you think about what
types of lines are not parallel, well, this green line
and this pink line are not parallel. They clearly intersect
at some point. So these two guys are parallel
right over here, and sometimes it's specified, sometimes
people will draw an arrow going in the same direction to show
that those two lines are parallel. If there are multiple parallel
lines, they might do two arrows and two arrows or whatever. But you just have to say
OK, these lines will never intersect. What we want to think about
is what happens when these parallel lines are
intersected by a third line. Let me draw the
third line here. So third line like this. And we call that, right there,
the third line that intersects the parallel lines we
call a transversal line. Because it tranverses
the two parallel lines. Now whenever you have a
transversal crossing parallel lines, you have an interesting
relationship between the angles form. Now this shows up on a lot
of standardized tests. It's kind of a core type
of a geometry problem. So it's a good thing to really
get clear in our heads. So the first thing to realize
is if these lines are parallel, we're going to assume these
lines are parallel, then we have corresponding angles
are going to be the same. What I mean by corresponding
angles are I guess you could think there are four angles
that get formed when this purple line or this
magenta line intersects this yellow line. You have this angle up here
that I've specified in green, you have -- let me do another
one in orange -- you have this angle right here in orange, you
have this angle right here in this other shade of green, and
then you have this angle right here -- right there
that I've made in that bluish-purplish color. So those are the four angles. So when we talk about
corresponding angles, we're talking about, for example,
this top right angle in green up here, that corresponds to
this top right angle in, what I can draw it in that same
green, right over here. These two angles
are corresponding. These two are corresponding
angles and they're going to be equal. These are equal angles. If this is -- I'll make up
a number -- if this is 70 degrees, then this angle
right here is also going to be 70 degrees. And if you just think about it,
or if you even play with toothpicks or something, and
you keep changing the direction of this transversal line,
you'll see that it actually looks like they should
always be equal. If I were to take -- let me
draw two other parallel lines, let me show maybe
a more extreme example. So if I have two other parallel
lines like that, and then let me make a transversal that
forms a smaller -- it's even a smaller angle here -- you see
that this angle right here looks the same as that angle. Those are corresponding angles
and they will be equivalent. From this perspective it's kind
of the top right angle and each intersection is the same. Now the same is true of the
other corresponding angles. This angle right here in this
example, it's the top left angle will be the same as the
top left angle right over here. This bottom left angle will
be the same down here. If this right here is 70
degrees, then this down here will also be 70 degrees. And then finally, of course,
this angle and this angle will also be the same. So corresponding angles -- let
me write these -- these are corresponding angles
are congruent. Corresponding angles are equal. And that and that are
corresponding, that and that, that and that,
and that and that. Now, the next set of equal
angles to realize are sometimes they're called vertical angles,
sometimes they're called opposite angles. But if you take this angle
right here, the angle that is vertical to it or is opposite
as you go right across the point of intersection is this
angle right here, and that is going to be the same thing. So we could say opposite -- I
like opposite because it's not always in the vertical
direction, sometimes it's in the horizontal direction, but
sometimes they're referred to as vertical angles. Opposite or vertical
angles are also equal. So if that's 70 degrees, then
this is also 70 degrees. And if this is 70 degrees,
then this right here is also 70 degrees. So it's interesting, if that's
70 degrees and that's 70 degrees, and if this is 70
degrees and that is also 70 degrees, so no matter what this
is, this will also be the same thing because this is
the same as that, that is the same as that. Now, the last one that you need
to I guess kind of realize are the relationship between
this orange angle and this green angle right there. You can see that when you add
up the angles, you go halfway around a circle, right? If you start here you do
the green angle, then you do the orange angle. You go halfway around the
circle, and that'll give you, it'll get you to 180 degrees. So this green and orange angle
have to add up to 180 degrees or they are supplementary. And we've done other videos on
supplementary, but you just have to realize they form the
same line or a half circle. So if this right here is 70
degrees, then this orange angle right here is 110 degrees,
because they add up to 180. Now, if this character right
here is 110 degrees, what do we know about this
character right here? Well, this character is
opposite or vertical to the 110 degrees so
it's also 110 degrees. We also know since this angle
corresponds with this angle, this angle will also
be 110 degrees. Or we could have said that
look, because this is 70 and this guy is supplementary,
these guys have to add up to 180 so you could have
gotten it that way. And you could also figure out
that since this is 110, this is a corresponding angle,
it is also going to be 110. Or you could have said
this is opposite to that so they're equal. Or you could have said that
this is supplementary with that angle, so 70 plus
110 have to be 180. Or you could have said 70
plus this angle are 180. So there's a bunch of ways
to come to figure out which angle is which. In the next video I'm just
going to do a bunch of examples just to show that if you know
one of these angles, you can really figure out
all of the angles.