Triangle angle example 2 | Angles and intersecting lines | Geometry | Khan Academy

Triangle angle example 2 | Angles and intersecting lines | Geometry | Khan Academy


We’re given a
bunch of lines here that intersect in all different
ways and form triangles. And what I want to
do in this video, we’ve been given the measures of
some of the angles, this angle, that angle, and that angle. And what we want
to do in this video is figure out what the
measure of this angle is. And we’re going to
call that measure x. And so I encourage you to pause
the video right now and try it yourself. And then I’m going to
give you the solution. So I’m assuming
you’ve unpaused it. And you’ve solved it
or you’ve given it at least a good shot of it. So let’s try to do it. And what’s fun about
these is there’s multiple ways to solve these. And you kind of
just have to keep figuring out what
you can figure out. So let’s say you start on the
left-hand side right over here. If this is 121 degrees,
then you’d say, well look, this angle right over here is
supplementary to this angle right over there. So this is 121 degrees
plus this green angle, that has to be equal to 180 degrees. So this is going to
be 180 minus 121. Let’s see, that’s the
same thing as 80 minus 21. 80 minus 20 would be 60. So that’s going
to be 59 degrees. So let me write that down. That’s going to be 59 degrees. Now we see that we have
two angles of a triangle. If you have two
angles of a triangle, you can figure out
the third angle, because they need
to add up to 180. Or you could say that this
angle right over here– so we’ll call that
question mark– we know that 59 plus
29 plus question mark needs to be equal
to 180 degrees. And if we subtract the 15 out
of the 29 from both sides, we get question mark is equal to
180 minus 59 minus 29 degrees. So that is going to be 180
minus 59 minus 29, let’s see, 180 minus 59, we
already know, is 121. And then 121 minus 29. So if you subtract
just 20, you get 101. You subtract another
9, you get 92. So that’s going to be
equal to 92 degrees. This is equal to 92 degrees. Well, this right here
is equal to 92 degrees. This angle right here is
vertical with that angle. So it is also going to
be equal to 92 degrees. And now we’re
getting pretty close. We can zoom in on this
triangle down here. And let me save some space here. So let me just say
that that over there is also going to be 92 degrees. And at this triangle
down here, we have two of the sides
of the triangle. We just have to
figure out the third. And actually, we don’t even
have to do much math here, because we have two of the
angles of this triangle. We have to figure
out the third angle. So over here, we have one angle
that’s 92, one angle that’s 29. The other one will be
180 minus 92 minus 29. And we don’t even have
to do any math here, because essentially, this is the
exact same angles that we have in this triangle
right over here. We have a 92 degree angle,
we have a 29 degree angle, and the other one is 59 degrees. So in this case,
it has to be also 59 degrees, because over
here they added up to 180. So over here, they’ll
also add up to 180. So that will also
get us to 59 degrees. We could just get that by
taking 180, subtracting 29, subtracting 92. And then if this is 59
degrees, then this angle is also going to be 59
degrees, because they are vertical angles. So we’re done. x is
equal to 59 degrees. Now there’s multiple
ways that you could have reasoned
through this problem. You could have immediately
said– so let me start over, actually. Maybe a faster way,
but you wouldn’t have been able to do kind
of this basic steps there, is you said, look, this is an
exterior angle right over here. It is equal to the sum of
the remote interior angles. So 121 is going to be 29 plus
this thing right over here. And we ended up doing that when
I did it step-by-step before. But here, we’re just
using kind of a few things that we know about
triangles ahead of time to maybe skip a step or two. Although I like to do it the
other way just so we make sure we don’t do anything weird. So anyway, this
is going to be 129 minus 29, which
is going to be 92. And if this is 92, then
this is also going to be 92. And then, if this is x, then
this is also going to be x. And you could say x plus 92
plus 29 is equal to 180 degrees. And then you’d say
x plus 92 plus 29 is going to be 121 degrees. We already knew that before. And so that is going
to equal 180 degrees. And so x is equal to 59 degrees. So there’s a ton
of ways that you could have thought
about this problem.

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