Lesson 1: Relationships of angles
Let's say we have one ray over here that starts at point A and then goes through point B. And so we could call this ray-- let me draw that a little bit straighter-- we could call this ray AB. Ray AB starts at A, or has a vertex at A. And let's say that there's also a ray AC. So let's say that C is sitting right over there. And then I can draw another ray that goes through C. So this is ray AC. And what's interesting about these two rays is that they have the exact same vertex. They have the exact same vertex at A. And in general, what we have when we have two rays that have the exact same vertex, you have an angle. And you're probably already reasonably familiar with the concept of an angle, which I believe comes from the Latin for corner, which makes sense. This looks like a little bit of a corner right over here that we see at point A. But the geometric definition, or the one that you're most likely to see, is when two rays share a common vertex. And that common vertex is actually called the vertex of the angle. So A is the vertex. Not only is it the vertex of each of these rays, ray AB and ray AC, it is also the vertex of the angle. So the next thing I want to think about is how do we label an angle. You might be tempted to just label it angle A. But I'll show you in a second why that's not going to be so clear to someone, based on where our angle is actually sitting. So the way that you specify an angle-- and hopefully this will make sense in a second-- is that you say angle-- this is the symbol for angle, and it actually looks strangely similar to this angle right over here. But this little pointy thing, or it almost looks like a less-than sign. But it's not quite. It's flat on the bottom right over here. This is the symbol for angle. You'd say angle BAC. Or you could say angle CAB. In either case, they're kind of specifying this corner. Or sometimes you could view it as this opening right over here. And the important thing to realize is that you have the vertex in the middle of the letters. Now you might be saying, wait, why go through the trouble of listing all three of these letters. Why can't I just call this angle A? And to see that, let me show you another diagram. And although the geometric definition of an angle involves two rays that have the same vertex, in practice, you're going to see many angles that are made up of lines and line segments. And you could imagine that you could continue those line segments on and on in one direction. And then they would become rays. So in that way, they're consistent with this definition. But let's say I have one line segment that looks like that. Let me label some points here. So we've already used ABC. So I'm going to call this D and E, points D and E. So this is line segment DE. And let's say I also have a line segment FG. And let's say this point where these two line segments intersect, let's call that point point H. Now how could we specify this angle right over here? Can we just call that angle H? Well, no. Because if we just said angle H, the angle that has a vertex H, it could be this angle right over here. Or it could be this angle right over here. Let me draw it this way. You could view it that way. Or it could be that angle over there. It could be this angle over here. It could be this angle over here. Or it could be that angle over there. And so the only way to really specify which angle you're talking about well, is to give three letters. So if you really did want to talk about that angle right over there, you would call that angle EHG. So that is angle EHG. Or you could actually call that angle GHE. If you wanted to specify this angle right over here, the one made up of, if you imagine that ray and that ray, if you were to keep on going past those points, then you could call that angle DHG, or angle GHD. I think you get the point. This angle up here could be FHD or EHF. And this one could be FHD or DHF. And when you do it this way, it's very clear what angle you are referring to. So now that we have a general idea of what an angle is, and kind of how do we denote it with symbols, the next thing you might be curious about is, it doesn't look like all angles are kind of the same. It seems like some angles open up or are more open than others. And some are a little bit more closed in than others. And that actually is the case. So for example, let's take two angles here. So let's say I have one angle that looks like that. So I'll started reusing letters. So let's say that this is A, B and C. I could make these rays. I could keep on going and make them rays if I like. Or I could just keep them as line segments. So right over here, I have angle BAC. And let's say over here, I have angle-- so let me draw another one-- and let's say this is angle XYZ. And once again, I could draw them as rays if I like, to go on and on and on. So it's angle XY and Z. And so when you just look at these, you just eyeball these two angles, it looks like this one is more open. So this one looks more open. While this one over here looks more closed, at least relative to this one. So maybe when we measure angles, we should measure it based on how open or closed they are. And that actually is the case. And so without even telling you how we measure an angle, you could say that the measure of angle XYZ, the measure of this angle, is greater than the measure of this angle right over here. And any convention we use for measuring angles is essentially going to be a measure of how open or how closed an angle actually is. And I'll take that up in the next video where we'll see how to actually measure an angle.