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## Geometry (all content)

# Angles (part 2)

More on complementary and supplementary angles. Introduction to opposite angles. Created by Sal Khan.

## Video transcript

So let's review everything that
we know so far, because it's good to keep reviewing. Because these are things
you should never forget the rest of your life. So if I have a line and if I
draw an angle that goes-- let's say this is the
pivot point, right? If I go all the way around
the line, or in a circle, that's 360 degrees. We learned that there are
360 degrees in a circle. Right? We also learned that if
I have lines like this. If I have two angles-- let
me draw it like that. And this is angle x. This is angle y. x and y are supplementary. And that just means that
they add up to 180 degrees. x plus y is equal
to 180 degrees. And why does that make sense? Because look, if we add
up x plus y we have gone halfway around the circle. So that's 180 degrees, right? So this is part of the way, and
this is the rest of the way. So x plus y are going
to equal 180 degrees. So hopefully we
have learned that. And then let me switch colors
for the sake of variety. Let me use my line tool. If I have-- let's see,
I'm going to draw perpendicular lines. If I have that line, and
then I have that line. And they are perpendicular. And then I have another line. Let's say it goes like that. And then I say that
this is angle x. Woops. This is angle x. And this is angle y. Well, I said this line and this
line are perpendicular, right? So that means that they
intersect at a 90 degree angle. So we know that this whole
thing is 90 degrees. And so what do we
know about x plus y? Well, x plus y is going
to equal 90 degrees. Or we could say that x
and y are complementary. And I always get confused
between supplementary and complementary. You just got to memorize it. I don't know if there's
any-- let's see, is there any easy way? 180, supplementary. You could say that 180-- 100
starts with an O, which supplementary does
not start with. So there. There's your mnemonic. Complementary. And 90 starts with an N,
and complementary does not start with an N. That's your other mnemonic. Complementary. I don't know if I'm
spelling it right. Who cares? Let's move on. So let's learn some more
stuff about angles. And what I'm going to do is I'm
going to give you an arsenal, and then once you have that
arsenal you can just tackle these beastly problems that
I'm going to throw at you. So just take these for granted
right now, and then in a few videos, probably, we're
going to tackle some beastly problems. And you know, I'm
using variables here. And if you're not familiar
with variables you can put numbers here. If x was 30 degrees, then y
is going to be 60 degrees. Right? Or in this case, if x is, I
don't know, 45 degrees, then y is going to be 135 degrees. That other way. Let me draw another property of
angles of intersecting lines. So if I have two angles, two
lines that intersect like this. So a couple of
interesting things. So first, I'm going to teach
you about opposite angles. Let me switch colors. Let me switch to yellow. So if this is x degrees, then
it turns out that the angle opposite to it is also
equal to x degrees. And you don't believe me? Well let me prove it to you. Let's say we call this,
I don't know, let's call this y degrees. Right? And I'm going to prove
to you that the x and the y are the same. Well what do we know already? Let's call this other angle--
and I'm doing this to confuse you-- angle z. Well what do we know about
angle x and angle z? It may not be obvious to you
because I've drawn it slightly different, but I'll give you
a small hint with an appropriately
interesting color. So what angle is this
whole thing right here? Well I'm just going
along a line, right? That's halfway around a circle. So what angle is that? Well that's 180 degrees. So what does x plus z equal? Well, x plus z is going to
equal that larger angle. x plus purple z is going to
equal-- I think I'll switch to the blue; maybe it's taking too
much time for me to switch-- is equal to 180 degrees. Or x and z are supplementary. I've run out of space. So what do we know about z? Well z is equal to 180 minus x. Right? Because x plus z is 180. Fine. Now, what's the relationship
between z and y? Well, z and y are
also supplementary. Because look, if I
drew this angle here. Look at this big angle. What angle is that? Well once again I'm still going
halfway around the circle. Right? But now I'm using this
line right here. So that's 180 degrees. So we know that angle z
plus angle y is also equal to 180 degrees. Right? Or, I don't want to keep
writing it, but z and y are also supplementary. But we just figured out
that z is 180 minus x. Right? So let's just substitute
that back in here. So we get 180 minus x plus
y is equal to 180 degrees. Why don't we subtract 180
degrees from both sides of this equation. That cancels out, and we get
minus x plus y is equal to 0. And then add x to both sides
of this equation, and we get y is equal to x. That was a very long way of
showing you something that is fairly simple-- that opposite
angles are equal to each other. So x is equal to y. And if you've played around
with this, if you just drew a bunch of straight lines and
they intersected at different angles, I think when you
eyeball it it would make sense. And then similarly, if that's z
then the other opposite angle here is also z degrees. So what do we know now? The total angles in a
circle, 360 degrees. When two angles kind of
combine, go halfway around the circle-- or they combine,
kind of form a line. There's different ways
you can think about it. We know they're supplementary. They add up to 180 degrees. x plus y is 180 degrees. If they add up to 90
it's complementary. x plus y is 90. And then opposite angles
are equal to each other. Right? This angle is equal
to this angle. And then this angle is going to
be equal to this angle for the same reason-- because
it's opposite. In the next video I'm going
to show you about parallel lines and transversals. More fancy words for what
I think are fairly straightforward concepts. I'll see in the next video.