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# Angle measurement & circle arcs

Video transcript

We already know that
an angle is formed when two rays share
a common endpoint. So, for example, let's say
that this is one ray right over here, and then this is one
another ray right over here, and then they would
form an angle. And at this point
right over here, their common endpoint is called
the vertex of that angle. Now, we also know that not
all angles seem the same. For example, this
is one angle here, and then we could
have another angle that looks something like this. And viewed this way,
it looks like this one is much more open. So I'll say more open. And this one right over
here seems less open. So to avoid having to just say,
oh, more open and less open and actually becoming a little
bit more exact about it, we'd actually want to
measure how open an angle is, or we'd want to have a
measure of the angle. Now, the most typical way
that angles are measured, there's actually two major
ways of that they're measured. The most typical
unit is in degrees, but later on in
high school, you'll also see the unit of radians
being used, especially when you learn trigonometry. But the degrees convention
really comes from a circle. So let's draw ourselves
a circle right over here, so that's a circle. And the convention is that--
when I say convention, it's just kind of what
everyone has been doing. The convention is that you
have 360 degrees in a circle. So let me explain that. So if that's the
center of the circle, and if we make this ray our
starting point or one side of our angle, if you go all
the way around the circle, that represents 360 degrees. And the notation is 360, and
then this little superscript circle represents degrees. This could be read
as 360 degrees. Now, you might be saying, where
did this 360 number come from? And no one knows for sure,
but there's hints in history, and there's hints in just the
way that the universe works, or at least the Earth's
rotation around the sun. You might recognize
or you might already realize that there are 365
days in a non-leap year, 366 in a leap year. And so you can imagine ancient
astronomers might have said, well, you know, that's
pretty close to 360. And in fact, several
ancient calendars, including the Persians
and the Mayans, had 360 days in their year. And 360 is also a much
neater number than 365. It has many, many more factors. It's another way of saying it's
divisible by a bunch of things. But anyway, this has just been
the convention, once again, what history has handed
us, that a circle is viewed to have 360 degrees. And so one way we
could measure an angle is you could put one of the
rays of an angle right over here at this part of the circle, and
then the other ray of the angle will look something like this. And then the fraction of
the circle circumference that is intersected by these two
rays, the measure of this angle would be that
fraction of degrees. So, for example, let's say that
this length right over here is 1/6 of the circle's
circumference. So it's 1/6 of the
way around the circle. Then this angle
right over here is going to be 1/6 of 360 degrees. So in this case, this
would be 60 degrees. I could do another example. So let's say I had a circle like
this, and I'll draw an angle. I'll put the vertex at
the center of the angle. I'll put one of the
rays right over here. You could consider
that to be 0 degrees. Or if the other ray was also
here, it would be 0 degrees. And then I'll make the
other ray of this angle, let's say it went straight up. Let's say it went
straight up like this. Well, in this
situation, the arc that connects these two
endpoints just like this, this represents 1/4 of the
circumference of the circle. This is, right over here,
1/4 of the circumference. So this angle right over here is
going to be 1/4 of 360 degrees. 360 degrees divided by 4
is going to be 90 degrees. At an angle like this, one where
one ray is straight up and down and the other one goes to
the right/left direction, we would say these two
rays are perpendicular, or we would call
this a right angle. And the way that we
oftentimes will denote that is by a symbol like this. But this literally
means a 90-degree angle. Let's do one more example. Let's do one more
example of this, just to make sure that we
understand what's going on. Actually, at least
one more example. Maybe one more if we have time. So let's say that we have an
angle that looks like this. Once more, I'm going
to put its vertex at the center of the circle. That's one ray of the angle. And let's say that
this is the other ray. This right over here is
the other ray of the angle. I encourage you to
pause this video and try to figure out what
the measure of this angle right over here is. Well, let's think about where
the rays intersect the circle. They intersect there and there. The arc that connects
them on the circle is that arc right over there. That is literally half of the
circumference of the circle. That is half of the
circumference, half of the way around of the circle,
circumference of the circle. So this angle is going to
be half of 360 degrees. And half of 360 is 180 degrees. And when you view it
this way, these two rays share a common endpoint. And together, they're
really forming a line here. And let's just do
one more example, because I said I would. Let me paste another circle. Let me draw another angle. Let me draw another angle. So let's say that's
one ray of the angle, and this is the other ray. This is the other ray of
the angle right over here. And we care. There's actually two
angles that are formed. There's actually two angles
formed in all of these. There's one angle that's
formed right over here, and you might recognize that
to be a 90-degree angle. But what we really care
about in this example is this angle right over here. So once again, where does
it intersect the circle? We care about this
arc right over here, because that's the
arc that corresponds to this angle right over here. And it looks like we've
gone 3/4 around the circle. So this angle is going
to be 3/4 of 360 degrees. 1/4 of 360 degrees is
90, so three of those is going to be 270 degrees.