 A proper understanding of angles is necessary to understand trigonometry. An angle is a thing, and, well, I know what it is when I see it. More importantly, let's talk about the parts of an angle. The vertex of the angle is the point where these two lines meet, and the legs of the angle are the rays leading away from the vertex. How do we measure angles? Well, suppose we have two angles like this. Obviously, this is the bigger angle. And not so fast. The problem is that the rays that make up the legs of the angle can be extended indefinitely. And what this means is that measuring an angle is problematic. So we define it this way. The measure of an angle is the amount of rotation required to turn one leg into the other about the vertex. A counterclockwise rotation is considered a positive measure, and a clockwise rotation is considered a negative measure. So we might take one leg of the angle and rotate it around the vertex until we get to the other leg. Since we rotated counterclockwise, this would give us a positive measure. We could also rotate clockwise, and the amount of rotation would be a negative amount. Now in order to measure the amount of rotation, there are three common units of angle measure. The most natural is the turn. If you tell someone to turn halfway around, or to turn all the way around, or to turn around five times, they'll know what you mean. While the turn is the most natural unit of measure, the most useful is something called the radian. We'll talk about radians in more detail later. But of all the ways to measure angles, the worst, least useful, most inconvenient is the degree. Probably the most common appearance of angles in everyday life is on the hands of a clock. Not that type of clock. A more traditional, old-timey clock. The hour and minute hand of a clock form an angle. So let's try to find the angle between the hands at three o'clock, at five o'clock, at ten o'clock, as measured from the minute hand, that's the longer of the two hands, to the hour hand, that's the shorter. Since we want to solve geometry problems in the hardest way possible, we don't draw a picture. Wait, wait, that's the wrong script. Since we want to solve geometry problems in the easiest way possible, we draw a picture. So we'll draw a picture. And remember, the measure of an angle is determined by the amount of rotation to turn one leg into the other around the vertex. So we want to rotate from the minute hand to the hour hand. So if we do that rotation, we see that at three o'clock, we need to make a quarter turn clockwise to rotate from the minute hand to the hour hand. And since we rotated clockwise, it's a negative measure. This means that the angle is minus one quarter turn. What if we go to five o'clock? Maybe we can speed things up a bit. So let's rotate from the minute hand to the hour hand. And because the clock is divided into twelves, we see that we need to rotate five twelves of a turn clockwise to rotate from the minute hand to the hour hand. And since this was a clockwise rotation, the angle measure is negative. This means the angle is minus five twelves of a turn. So let's go to ten o'clock. Now remember, we're rotating from the minute hand to the hour hand. So that at ten o'clock, we need to rotate two twelves of a turn counterclockwise to rotate from the minute hand to the hour hand. Since this was a counterclockwise rotation, the angle measure is positive, and this means the angle is two twelves of a turn. Now for an o'clock, we can keep going around. For example, in 24 hours, the hour hand of a clock goes around twice clockwise. So it marks out an angle of minus two turns. Two, because it's two full turns, and minus for the clockwise rotation. So let's talk about an unnatural measure of an angle. The problem is that if we use turns to measure angles, most angles will have fractional measures. And I know everybody loves fractions and wishes every problem would have nothing but fractions in it. Well, actually most of us don't like fractions. And when we don't like something, we often react by doing something illogical and against our better interests. So we might vote for a populist demagogue with delusions of grandeur. Or in this case, we might define something called a degree and say that one turn is 360 degrees. And this will help us avoid fractions. So for example, suppose we have an angle with a measure of 3 tenths of a turn. What's our degree measure? So we have 3 tenths of a turn. But since one turn is 360 degrees, then any time I see a turn, I can replace it with 360 degrees. So I have 3 tenths of 360 degrees. And that's really 3 tenths of 360, which gives us 108 degrees. Or maybe I have an angle with a measure of minus 3 turns. So again, we have minus 3 turns. And since a turn is 360 degrees, I can replace and calculate.