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# Worked example: Triangle angles (diagram)

Video transcript

So in this diagram over here,
I have this big triangle. And then I have all these
other little triangles inside of this big triangle. And what I want to do is
see if I can figure out the measure of this
angle right here. And we'll call
that measure theta. And they tell us a
few other things. You might have seen
this symbol before. That means that these
are right angles or that they have a
measure of 90 degrees. So that's a 90-degree angle,
that is a 90-degree angle, and that is a 90-degree
angle over there. And they also tell us that this
angle over here is 32 degrees. So let's see what we can do. And maybe we can solve this
in multiple different ways. That's what's really
fun about these is there's multiple ways
to solve these problems. So if this angle
is theta, we have theta is adjacent
to this green angle. And if you add them
together, you're going to get this right angle. So this pink angle, theta,
plus this green angle must be equal to 90 degrees. When you combine them,
you get a right angle. So you could call this
one-- its measure is going to be 90 minus beta. And now we have three
angles in the triangle, and we just have
to solve for theta. Because we know this angle
plus this angle plus this angle are going to be
equal to 180 degrees. So you have 90 minus theta plus
90 degrees plus 32 degrees-- so I'm going to do that
in a different color-- is going to be equal
to 180 degrees. The sum of the measures of
the angle inside of a triangle add up to 180 degrees. That's all we're
doing over here. And so let's see if we can
simplify this a little bit. So these two guys--
90 plus 90's going to be 180, so you get
180 minus theta plus 32 is equal to 180 degrees. And then what else do we have? We have 180 on both sides. We can subtract that
from both sides. So that cancels out. That goes to 0. And then you have negative theta
plus 32 degrees is equal to 0. You can add theta to both sides. And you get 32 degrees
is equal to theta, or theta is equal to 32 degrees. So it's going to actually be
the same measure as this angle right over here. That's one way to
do the problem. There's other ways that we
could have done the problem. Actually, there's a ton of
ways we could have done this. We could have looked at
this big triangle over here. And we could've said, look. If this is 90 degrees over here,
this is 32 degrees over here, this angle up here
is going to be 180 minus 90 degrees
minus 32 degrees. Because they all have to
add up to 180 degrees. And I just kind of
skipped a step there. Actually, let me
not skip a step. Let me call this x. If we call the measure
of that angle x, we would have x plus 90. I'm looking at the biggest
triangle in this diagram right here. x plus 90 plus 32 is going
to be equal to 180 degrees. And so if you subtract 90
and 32 from both sides. So if you subtract 90 from
both sides, you get x plus 32 is equal to 90. And then if you subtract
32 from both sides, you get x is equal to--
what is this-- 58 degrees. Fair enough. Now, what else
can we figure out? Well, if this angle over here
is a right angle-- and I'm just redoing the problem
over again just to show you that there's multiple
ways to get the answer. We were given that
this is a right angle. If that is 90 degrees,
then this angle over here is supplementary to it, and
it also has to be 90 degrees. So then we have this angle
plus 90 degrees plus this angle have to equal 180. Maybe we could call that y. So y plus 58 plus
90 is equal to 180. You can subtract
90 from both sides. Subtract 90 from both sides. This will become 90. Subtract 58 from both sides, you
get y is equal to 32 degrees. Well, if y is 32 degrees,
it is complementary. It is complementary to
this angle right over here. It is complementary-- let
me do that in a new color, not supplementary. It is complementary. It adds up to 90 degrees. It is complementary to
this angle over here. We could call it z. So these two combined are
going to add up to 90 degrees, or z is going to be
equal to 58 degrees. And now we're
inside the triangle that we care about
to figure out theta, theta that we've already figured
out earlier in this video. Well, this z is 58 degrees. If this angle over here is
90, then this one over here is also going to be 90,
because they're supplementary. So you have 58 degrees. I wanted to do that
in that orange color. So if you have 58
degrees, so you have 58 plus this 90,
plus 90, plus theta now is going to be
equal to 180 degrees. You can subtract
90 from both sides. That becomes 90,
and then you have 58 plus theta is equal to 90. Subtract 58 from both sides. You get theta is equal
to 32 degrees again. And so we've got
the same answer. I just wanted to
do that, show you that there are multiple
ways to do these problems. And as long as you're doing
things that are logically consistent, you're making
assumptions that you can make and then logically
deducing step by step, there's multiple ways to
get that right answer.