Measuring angles in degrees

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.