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## Geometry (all content)

# The angle game (part 2)

More examples of the Angle Game. Created by Sal Khan.

## Video transcript

Welcome back. Let's do a couple more angle
game problems, and hopefully this will make you an
angle game expert. So let's start, I have the star
drawn again, and let's say we know the following angles. We know this angle right
here is 41 degrees. We know this angle
here is 113 degrees. We know this angle
here is 101 degrees. And what we have to figure out
-- this is the goal of this angle game -- we want to figure
out what this angle is. And like always, I encourage
you to try it on your own. Pause the video and then just
try to work it through. If you get stuck, then play
the video again and hopefully I'll have a solution for you. So pause right now, but
otherwise let me explain how to do this. So let's see, we know this,
this and this, and we're going to figure out this angle. So how can we figure
out this angle? What are the possible
strategies? Well, if we knew this
angle here, we could say they're supplementary. But that angle seems like a
hard angle to figure out too, because it's not a
part of any triangles. But this angle is a
part of this triangle right here, right? So if we were able to figure
out this angle and this angle, these green angles, if we're
able to figure out these green angles, then we could figure
out this brown angle, which is the goal of this angle game. So, this could also be a good
time to pause because I just gave you a hint. This green angle, well it's
supplementary to this angle right here, so that means it
adds up to 180 degrees, and that's clear because it's
on kind of the same line. So this is 101 degrees
and this is going to be 79 degrees, right? So it adds up to 180 degrees. That's 79. Now how can we figure
out this angle? Well, it's kind of left by
itself out in the corner of some place, so we could see if
it's part of any triangles. But we already said it's
part of this triangle. But that doesn't help us
because we don't know this angle and that's
actually our goal. What other triangles
is it a part of? Well, it's a part of this
triangle right here. That's why I like the star
problem because it has all these triangles in it that
might not be obvious to you the first time you look at it. But the more you look at you
see all these triangles. So it's part of this
triangle, and it's also part of this triangle. I'm going to draw this triangle
another color because I think it'll be clear to you that this
is a useful triangle to see that's it's a part of. So we have that triangle. So do we know two of the
angles of that triangle? Well sure. We know this angle and
we know this angle. So we know that this angle plus
113 plus 41 is going to equal 180 degrees because of the
three angles of a triangle. So let me call this, I
don't know, g for green. Let's call this g for green. So we know g plus 113 degrees,
that's this one right here, plus 41 -- remember, we're
looking at this triangle; that's the hardest part just
keeping track of which triangle we're looking at -- is
going to equal 180 degrees. g plus, what is this, 154? Right? 40, 50, 154 equals 180 degrees. That's always where I
mess up on the addition. And so g is equal to, what is
this, 26 degrees, right, because I just subtract
154 from both sides. So we're almost there. So we figured out g, we
know this green angle. We just have to figure out
this, and they're all part of this triangle, this
small one right here. This small triangle. So our goal, which is
let's call this x. x plus g, which is 26 degrees
-- we just figured that out. 26 plus this angle, 79 -- and
we figured that out because it was supplementary to this
angle -- is going to equal 180 degrees. So x plus, what is this,
105 equal to 180. So x is equal to 75 degrees,
if I did my addition and subtraction correctly. So x is equal to 75 degrees. And then we are done. Let's do another one
of these problems. These problems are all
generated on the [? Card ?] Academy website dynamically
by the computer. Whoever wrote this software
must be a genius. But anyway, back
to the problem. Let me draw some more. So this is going to be a pretty
straightforward drawing. It's pretty much just two
triangles next to each other. Like that and then let me draw
another line that goes like that, and then we draw a line
that goes like that, and I think I have done my drawing. There you go. I'm have done my drawing. So let's see. What do we know about this
triangle and what do we need to figure out? I'm going to tell you that this
angle here, this big angle here, is 86 degrees. We also know that this
angle here is 28 degrees. And we also know that this
angle here is 122 degrees. And our goal, our mission
in this round is to figure out what this angle is. And maybe we can do it, we
can do it in a good color. Maybe we can do it in a
couple of different ways. So one thing we could do is we
could figure out what this angle is, so we could just
subtract this green angle from 86 and we would get our answer. Well, this angle's easy, right,
because we know two angles of this triangle, so we
could figure that out. Let's just call this, I don't
know, let's call this y. So y plus 122 plus 28 degrees
is going to equal 180. So y plus 150 is equal to 180. So y is equal to 30
degrees, right? So this is equal to 30 degrees. So this is 30 degrees, and
this big angle here is 86. So our goal, let's call that x,
so x is going to just be equal to the big angle, 86 minus this
angle we just figured out, minus 30. So x is going to be
equal to 50 degrees. Done. That was a pretty
straightforward problem. Let's see if we could figure
that out any other way. Well, we could say instead
of doing it that way -- let's forget we just
solved it that way. We could say this angle here
is supplementary to this 122 degree angle, right, so
it has to add up to 180. So this plus 122 is 180, so
what does that make this? It makes this 58
degrees, right? This plus this is
going to be 180. So we figured out this. If we could figure out
this, then we could use this triangle. How do we figure
out this angle? Well, we could look at this big
triangle here, and we know this side, right, and we
could figure out this. Let's call this z. So we know that z plus this
angle, plus 28, plus this big angle, plus 86 is equal to 180. So z plus, what is this,
106, 114 is equal to 180. So z is equal to, what
is this, 66 degrees. I don't know if I'm doing
any of my math correctly, but let's just hope. z equals 66. So z is 66, this angle is 58,
and now we can use this triangle here to figure out
what this angle is, our x. So x plus 66 plus 58
is equal to 180. I already think I might
have made a mistake some place in the addition. So this time around I get x
is equal to -- let's see, 66 plus 58 is 110 plus 14. So 180 minus 124. So now I got it, x is
equal to 56 degrees. Oh great, I actually
got the right answer. I was looking at this, I
thought it was 50, but this was 56, right -- 86 minus 30. So x is equal to
56 degrees again. So we did it two
different ways. That's what I wanted
to show you. There's actually not a right
answer, as long as you kind of get there eventually. We solved it two different ways
and I did all my addition and subtraction correctly, and you
get the exact same answer. So hopefully you find the angle
game fun and you'll be playing this with your friends. I'll see you later.