 Welcome back to our lecture series, Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misalign. In this lecture 12, we're going to introduce the idea of what an angle means in an order geometry. While incidence geometry was too course to be able to give us notions like angles and triangles and things like that, between this exactly gives us what we need. Because we have things like plane separation, we can talk about what it means to be inside of a triangle or outside of a triangle. Ultimately, this is going to drive from our notion of an angle which we're going to define just right now. Consider three distinct non-colinear points in an order geometry. Then we can define the angle denoted as angle ABC by the following formula. The angle ABC is the intersection of the two half planes. It'll be the intersection of the half plane determined by the line AB and the point C, with the half plane determined by the line BC on the side of A. I want to draw a picture here to try to illustrate what's going on here. We have our three points. Maybe something like this is what we're thinking. We have our point A, we have our point B, and we have our point C. Then I want us to consider the ray from B to A and consider the ray from B to C. We know there's a line determined by these things. But when we think of the angle, we're really thinking of these two rays, but we also have everything inside the angle. I'll be more explicit about that in just a moment. The first part, you take the line AB, that of course is this line right here, and then there's two sides of the line that we've determined previously. Take the side that contains the point C. We're taking this side of the line right here, but then also consider the line BC, and it has two sides. There's this one and this one, and we want the side of the line that has the point A. That's going to be this side of the line right here. If we take the intersection of those two things, then the intersection of those half planes would be this area right here, and that gives us the so-called angle. I want to caution us that even though we've defined angles here, we don't have anything about angle measure. We can't talk about degrees. We can't talk about radians. We don't have things like distance and measure yet. That's something we'll develop later on in our lecture series, but at least at this moment we can talk about angles. So angles by definition are going to be the intersection of two half planes. In particular, they will contain the two rays that bound it. So you have your initial side right here that is part of the angle. You have your terminal side up over here that's also part of the angle. Then we have all the things that are between those two rays. And so that is what we define here as the angle. Now we can talk about the interior of an angle. The interior of an angle, which will be denoted, the angle symbol, then we're going to draw a little circle here. This is sort of a topological notation here, describing an open set here because we're no longer including the boundaries. So the interior of the angle or sometimes called the open angle, this will be the intersection of the open half planes. So remember, what's the difference between the half plane AB with respect to the point C and then the open half planes is the closed half plane. The closed half plane contains the boundary line itself while the open half plane doesn't contain the line. So that's the difference there. Same thing here. So when you look at the open angle, the interior of the angle, you don't get the rays BA or BC, but you do get everything that's else's there. So that's the interior of the angle. There are two special cases that we need to define right now. There's the example we call a half angle, excuse me, a flat angle. A flat angle is actually just a closed half plane. And that's cause when we defined an angle above, we said that the points AB and C were not collinear. Well, if AB and C are collinear, there's a couple of possibilities. You have that B is between A and C. And so it's basically like you have this ray and you have this ray with B between them. These are what we call opposite rays. We're gonna talk more about those in this video, well, not this video, but in this lecture, lecture 12 here. So you have this half angle, excuse me, this half plane. A half plane is what we call a flat angle. In the future, this is what would coincide with 180 degree angle. But again, we don't have measured yet, that's why I want you to think of it. This flat angle is just a half plane. Some people call these straight angles, that's a possibility. Now be aware that when you have a flat angle, you're choosing a side, you're not both of them simultaneous, you only have one of them, like so. You also have the notion of a null angle. A null angle happens when the ray BA coincides with the ray BC, right? So they're both on the same side going on right here. This is what we would normally think of as a zero degree angle, but again, this is sort of like a degenerate angle, much like a flat angle as well, because the points A, B and C are in fact collinear. This happens when you have the relationship that A is between B and C, or you have the relationship where C is between B and A. And all of these situations, flat angles and null angles as the name suggests, that we are gonna call them angles, but they're not angles proper. We have to always treat them exceptionally here. But when we talk about an angle, whether it's a genuine angle or one of these degenerate angles, this point B is somewhat more important than that, once it's often referred to as the vertex of the angle. And that's because it's the intersection of the two rays, BA and BC, that really are the boundary for this angle. And so with this definition of angle out of the way, we'll then proceed to prove some things about angles in order geometry.