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# Measuring angles in degrees

Video transcript

Now that we know
what an angle is, let's think about how
we can measure them. And we already hinted
at one way to think about the measure of angle in
the last video where we said, look, this angle XYZ seems
more open than angle BAC. So maybe the
measure of angle XYZ should be larger than
the angle of the BAC, and that is exactly
the way we think about the measures of angles. But what I want to
do in this video is come up with an exact
way to measure an angle. So what I've drawn over here is
a little bit of a half-circle, and it looks very
similar to a tool that you can buy at your
local school supplies store to measure angles. So this is actually a little bit
of a drawing of a protractor. And what we do in something
like a protractor-- you could even construct
one with a piece of paper-- is we've taken a
half-circle right here, and we've divided it
into a 180 sections, and each of these marks
marks 10 of those sections. And what you do
for any given angle is you put one of the
sides of the angle. So each of the rays
of an angle are considered one of its sides. So you put the
vertex of the angle at the center of
this half-circle-- or if you're dealing with
an actual protractor, at the center of
that protractor-- and then you put one
side along the 0 mark. So I'm going redraw this
angle right over here at the center of
this protractor. So if we said this is Y, then
the Z goes right over here. And then the other ray, ray
YX in this circumstance, will go roughly
in that direction. And so it is pointing on the
protractor to the-- let's see. This looks like this
is the 70th of section. This is the 80th section. So maybe this is, I would
guess, the 77th section. So this is pointing
to 77 right over here. Assuming that I drew it the
right way right over here, we could say the measure of
angle XYZ-- sometimes they'll just say angle XYZ is equal
to, but this is a little bit more formal-- the measure
of angle XYZ is equal to 77. Each of these little sections,
we call them "degrees." So it's equal to
77-- sometimes it's written like that, the same
way you would write "degrees" for the temperature outside. So you could write "77 degrees"
like that or you could actually write out the word
right over there. So each of these
sections are degrees, so we're measuring in degrees. And I want to be clear,
degrees aren't the only way to measure angles. Really, anything that
measures the openness. So when you go
into trigonometry, you'll learn that you
can measure angles, not only in degrees,
but also using something called "radians." But I'll leave that
to another day. So let's measure this
other angle, angle BAC. So once again, I'll
put A at the center, and then AC I'll put along the 0
degree edge of this half-circle or of this protractor. And then I'll point
AB in the-- well, assuming that I'm drawing
it exactly the way that it's over there. Normally, instead
of moving the angle, you could actually move the
protractor to the angle. So it looks something
like that, and you could see that it's pointing to
right about the 30 degree mark. So we could say that
the measure of angle BAC is equal to 30 degrees. And so you can look
just straight up from evaluating these
numbers that 77 degrees is clearly larger than 30
degrees, and so it is a larger angle, which makes sense
because it is a more open angle. And in general, there's a
couple of interesting angles to think about. If you have a 0 degree
angle, you actually have something that's
just a closed angled. It really is just a
ray at that point. As you get larger and larger or
as you get more and more open, you eventually get to a
point where one of the rays is completely
straight up and down while the other one
is left to right. So you could imagine
an angle that looks like this where one ray
goes straight up down like that and the other ray goes
straight right and left. Or you could imagine
something like an angle that looks like this where, at least,
the way you're looking at it, one doesn't look
straight up down or one does it look
straight right left. But if you rotate it, it would
look just like this thing right over here where one is
going straight up and down and one is going
straight right and left. And you can see from our
measure right over here that that gives us
a 90 degree angle. It's a very interesting angle. It shows up many, many times
in geometry and trigonometry, and there's a special word
for a 90 degree angle. It is called a "right angle." So this right over
here, assuming if you rotate it around,
would look just like this. We would call this
a "right angle." And there is a notation to
show that it's a right angle. You draw a little part of
a box right over there, and that tells us that this
is, if you were to rotate it, exactly up and down while
this is going exactly right and left, if
you were to rotate it properly, or vice versa. And then, as you
go even wider, you get wider and wider
and wider and wider until you get all the way to
an angle that looks like this. So you could imagine
an angle where the two rays in that
angle form a line. So let's say this is
point X. This is point Y, and this is point Z. You
could call this angle ZXY, but it's really
so open that it's formed an actual line here. Z, X, and Y are collinear. This is a 180 degree angle where
we see the measure of angle ZXY is 180 degrees. And you can actually
go beyond that. So if you were to go all
the way around the circle so that you would get back
to 360 degrees and then you could keep going around
and around and around, and you'll start to
see a lot more of that when you enter a
trigonometry class. Now, there's two
last things that I want to introduce in this video. There are special
words, and I'll talk about more types of
angles in the next video. But if an angle is
less than 90 degrees, so, for example, both of
these angles that we started our discussion with are
less than 90 degrees, we call them "acute angles." So this is acute. So that is an acute
angle, and that is an acute angle
right over here. They are less than 90 degrees. What does a non-acute
angle look like? And there's a word for
it other than non-acute. Well, it would be
more than 90 degrees. So, for example-- let
me do this in a color I haven't used-- an angle
that looks like this, and let me draw it a little
bit better than that. An angle it looks like this. So that's one side of the
angle or one of the rays and then I'll put the other
one on the baseline right over here. Clearly, this is
larger than 90 degrees. If I were to
approximate, let's see, that's 100, 110,
120, almost 130. So let's call that maybe
a 128-degree angle. We call this an "obtuse angle." The way I remember it as acute,
it's kind of "a cute" angle. It's nice and small. I believe acute in either Latin
or Greek or maybe both means something like "pin" or "sharp." So that's one way
to think about it. An acute angle
seems much sharper. Obtuse, I kind of
imagine something that's kind of
lumbering and large. Or you could think
it's not acute. It's not nice and
small and pointy. So that's one way
to think about it, but this is just
general terminology for different types of angles. Less than 90 degrees,
you have an acute angle. At 90 degrees, you
have a right angle. Larger than 90 degrees,
you have an obtuse angle. And then, if you get all
the way to 180 degrees, your angle actually
forms a line.