 The next major geometric topic is the topic of angles. So we'll introduce them in the following way. If I have two straight lines intersected a point, then what I form with those two straight lines are a set of angles. And the point of intersection of those lines is going to be what we call the vertex of the angle. And if you want to measure what the angle is, what we're going to do for that is we're going to measure it by the rotation of what we can think about as a direction arrow. So for example, consider two straight lines like this. And so now I have two straight lines. They're intersecting at a point that's going to be by vertex of the angle. And now I've created one, two, three, four distinct regions formed by the intersection of those two lines. And each of these regions is going to be an angle. All right, so let's try to measure one of these angles. Let's go ahead and measure this angle. And so what we'll do is we'll take a direction arrow and note that our direction arrow is initially pointing in the direction of this line. And the line points in two directions. It points this way. It could also point this way. But if we want to measure this angle here, we're going to start off by having the direction arrow point along the direction of one leg of this angle that's formed by the two line. Also notice one of the useful feature here. It's not really required, but it does make things a lot easier if we have one end of the direction arrow. So in this case, we have the end of the arrow at the vertex. And the tip of the arrow is going to be someplace along that line. We don't need that to happen, but it makes our lives a lot easier. Because what we're going to then do at this point is we're going to rotate that direction arrow so that it measures out the angle. And again, the amount of rotation is going to correspond to how much we have what the angle measure is going to be. So again, I'll do that rotation. So I'm going to rotate that direction arrow so that it includes all of the angle there. And if you want to, if you imagine there's a crowd of people out here where that direction arrow is pointing, everybody in the crowd is going to be pointed to at some point along that rotation. So there's our measure of the angle. However much we've rotated. Now we can consider a special case. Imagine that I have two straight lines meeting in such a way that all four angles that you form are equal. In that particular case, then we say that the two lines are perpendicular and the angles that we form are right angles. So for example, here I have two straight lines. They intersect at a point. Again, one, two, three, four regions, all of which are angles. And if it happens that these are, in some sense, equal, then what I have when the two lines meet is the two lines are perpendicular and the angles that we obtain are right angles. So the natural question we want to ask is, well, how big is one of these right angles? Well, let's consider what our rotations look like. So again, the measure of an angle is going to be based on how much of a rotation a direction arrow will make. So let's go ahead and put that direction arrow down there. And so this time I'm going to rotate that. And let's rotate that all the way around. So there's one complete rotation. And if I measure all four of those angles, then I've got to go through one complete turn. Now, this is actually going to be true for any pair of intersecting lines. What makes the case of a right angle different is that if we have a right angle, all four of those angles that we've just measured have the same measure. So if all four of those angles are equal, then all four of the angles are right angles. And so that means that if these are in fact equal, since I have a full turn that's distributed among four pieces, then each of these must represent one quarter of a full turn. So each of those right angles is a quarter turn, a quarter rotation of our direction arrow. Now that's worth keeping in mind. The natural unit of rotation is a turn. We can do a full turn. We can do a half turn. We know what these mean. They don't really require that much in the way of explanation. Somebody who has no idea what an angle is, you can still describe the idea of measuring a rotation turn halfway around. And so this is a very natural unit of rotation. The problem is that as an actual measure, it's a little bit too large for practical purposes. So we have to break the full turn up into smaller pieces. And so one very common way of doing that is to use the degree measurement. And that takes our full turn and breaks it up into 360 degrees. So 360 degrees corresponds to one full turn, which we can either take clockwise or counterclockwise. That would be part of the directions for how to do the turn. Now that's a fairly common unit, but there's a second unit that's actually more useful, and that's the radian. And the idea behind the radian measure is that again, I'm going to take my natural unit of rotation, the full turn, and I'm going to break that into a number of smaller units. And in this particular case, I'm going to break that into two pi radians. Well, the important thing here is that you know how much a full turn is. You can visualize what a full turn is. And if I break it into 360 degrees, or if I break it into two pi radians, it's easy to convert back from turns into degrees or into radians. So for example, a half turn, I'm going to go halfway around. Well, that's going to be half of 360 degrees. That's 180. Or maybe I want to express that in radians because it's a better unit. So it's going to be half of two pi radians, and that's just going to be a pi radians. Or maybe I could take a quarter turn. So again, remember our right angles. Those were all quarter turns. So how much is a quarter turn? Well, it's a quarter of 360 degrees. So it's 90 degrees. Or it's a quarter of two pi radians. It's going to be pi over two radians. And in general, we can find any angular measure that you want as long as you remember this equivalence. One full turn is identical to is equivalent to 360 degrees, is equivalent to two pi radians. And all of these things mean exactly the same thing, the natural unit of rotation, the best unit of rotation, and the one that we see most commonly.