Measuring angles in degrees How to measure an angle in degrees
Measuring angles in degrees
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- 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 an angle in the last video when 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 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 supply store to measure angles.
- So this is actually a little bit of a drawing of a protractor, and what we do with something like a protractor
- --and you can even construct one with a piece of paper--is we've taken a half circle right here,
- and we've divided it into 180 sections, and each of these marks marks ten 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
- the 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 zero mark. So I'm going to redraw this angle right over here
- at the center of this protractor, so we can say if this is "Y", than "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 it is the seventieth section,
- this is the eightieth section, so maybe this is, I would guess this is the 77th section,
- so this is pointing to right over here. So when we measure an angle--so we could say that the measure of XYZ
- --assuming I drew it the right way right over here--we could say the measure of
- angle XYZ--sometimes they'll say "angle XYZ is equal to" but this is a little bit more formal-- the
- measure of angle XYZ is equal to 77--and what we do is we call each of these little sections "degrees"--
- so it's equal to 77--sometimes it's written like that, the same way you would write degrees for a 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 in degrees so we're measuring in degrees, but 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. 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 this zero degree edge
- of this half circle or protractor, and then I'll put point AB in the--well, assuming I'm drawing it exactly
- the way that it is over there--normally instead of moving the angle you could actually move the protractor
- to the angle--it looks something like that. And you can see that it's pointing to right about--well let's
- just say to right about the 30 degree mark. So you could say the measure of angle BAC is equal to 30 degrees.
- And so you could 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 zero degree angle you have something that's just a closed angle--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-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 doesn't look striaght right-left, but if you rotate it, it would look just like this thing 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, which is a very
- interesting angle that shows up 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 we rotate it around it would look just like this--we would
- call this a right angle. And there is a notation to show that it is a right angle: you draw a little
- kind of part of a box right over there and that tells us that 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 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.
- The two rays--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 has formed an actual line here.
- Z,X, and Y are co-linear. And what we have over here, this is a 180 degree angle, or we should
- say 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'd get back to 360 degrees, and then you'd
- keep going round and round and round and you'll start to see a lot more of that when you enter
- a trigonometry class. Now there's one last--or two last things that I want to introduce in this video--
- is 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 anything, this is acute,
- 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--let me draw it a little bit better than that--an angle that looks like this,
- and that's one side of the angle, or one of the rays,
- and then I'll put the other one along the base-line 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 is 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, so that's how I remember it,
- or you can say 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 if you get all the way to 180 degrees, your angle actually forms a line.
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