 Welcome back to our lecture series math 1060 trigonometry for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In this, the first video for lecture two, we're going to be talking about the ideas of angles. In lecture one, we talked a lot about a triangle, which as you probably know, a triangle roughly translates as meaning three angles inside of a polygon. What is actually the angle part? That's what we want to emphasize in this lecture today. We're going to start this video with a little bit of vocabulary, a little bit of motivation. Imagine we have two points in the plane, so we have two points of column AB. As defined previously, if we draw the line segment that connects A and B, we would denote that as AB with a line over it. If we omit the line on top of it, that means the measure of that line segment, that is the distance between the points A and B. What we want to talk about right now though, is if we take the line that's connected by, that is the line that's determined by the points A and B, and so this is the line of infinite length that goes on forever and ever in both directions. This is going to be denoted as AB with a double-arrowed line on top of it. So the line determined by A and B. Given any two points, there's a unique line that contains those two points, and so we do say that two points determine a line. And as previously defined, we say that any set of points is collinear if there's a single line for which all points are incident to that single line. This is something we've talked about before as well. What I want to talk about now is next the idea of a ray. A ray is similar to that of a line. If we take the ray, and of course I'm not talking about ray skywalk or anything right here, this would be RAY. If we take the ray from A to B, and direction very much matters when it comes to rays, in fact, when we have a ray, we say that we have a vertex A, which is gonna be the first letter, the first point you list right here. And we would say that the ray emanates from the vertex A towards the point B. This ray is going to be all the points that are collinear to A and B, but in the direction of A and B. So basically what you mean by a ray is we wanna take all the points that are between A and B, right? But we also want all the points such that new point and A have B between them. So basically a ray is essentially half of a line. You take the line AB, but you go in the direction from A towards B. And so the ray emanating from A to B is not the same thing as the ray emanating from B to A or anything like that. These rays basically make half of a line. Now the reason we define that is that rays are how we build angles. So what is an angle? So an angle, well, it's gonna be determined by three points in the plane, which is called the ABC. And so we'll denote that angle as angle ABC. It's gonna be the span in the plane of two rays that share a common vertex. So let's say those rays are gonna be the ray BA and the ray BC. So the ray BA is the ray that emanates from the vertex B towards the point A and then ray BC is the ray that emanates from the point B towards some other point C inside the plane. As you can see, illustrated right here. So we have the ray BA, which looks like this right here. So it's all the points that emanate from B towards A and beyond A, of course. Then there's also the ray BC. So all the points that emanate from B towards C and beyond, we have these two rays. And so the angle would then include the first ray and includes the second ray. And it also includes all of the points in between those two rays. And so this is what one means by an angle. The first ray associated to the angle is called the initial side. And then the second ray that determines this angle is called the terminal side. And initial and terminal seem to have some type of temporal meaning that is there's some type of time. Like this is where we're going to start and the terminal side is where we're going to stop. I mean, after all, the city of Termina and Major's Mask is named from the fact that that city is going to come to an end when the moon crushes it, if you know what I mean. And the idea is that initial side, and we think in terms of rotation, the angle is spanned by rotating the initial side towards the terminal side. And so there is a direction associated to angles. And so this is what an angle is. An angle, in some regard, is a piece of a rotation. Like we talked about circles in a previous lecture, that is in lecture one. A circle, you think of it as one complete rotation. You go from, if you were to transverse the circumference of a circle, you go from start to stop. You get the whole thing. But what if we only cover some sector? Like what if we only covered one part of that circle? Not the whole circle, but it's a small sector of it. That coincides with a type of angle. It's part of a rotation. Now there are some special angles that we should refer to here. And so this is gonna be some angle vocabulary that's gonna be important for us going forward. We often will talk about a flat angle or a straight angle, which is essentially just a line, so to speak, which if you take a line, a line actually determines two flat angles. There's the one on top. There's the one on bottom. These are sometimes called half planes. That's another vocabulary we can use right here. Because after all, the angle is not just the rays that form it. It's also all the space in the plane encompassed in it as well. So a flat angle, like you would see right here, is an angle for which the initial side, so here's our vertex right here. You have the initial side. You have the terminal side over here. And these are co-linear with one another. So a flat angle is an angle where you have all three points are in fact co-linear, where the vertex B right here is between the initial point and the terminal point right here. So a flat angle. Similar to the idea of a flat angle is what we might call a null angle. A null angle also coincides with three points which are co-linear, but in the following manner, a null angle is just a single ray. So you have your one point A, your vertex B, but then the second ray, BC, is itself just the same ray as BA. So there's basically no space between the two rays because the two rays overlap with each other. Then continuing on here, we have the idea of supplementary angles. We say that two angles are supplementary if their union forms a flat angle. So for example, if you take one angle here in blue and then you take another angle here, say in yellow, these two angles are supplements to each other because the union of the two angles forms a half plane. That is they connect together to form this line. Similarly, we defined the idea of a right angle earlier. If two lines meet so that they're perpendicular with one another, that is if you have two perpendicular lines, we say that the angle form between them is a right angle. Another one, and we define that in terms of slope, two lines are perpendicular if the product of their slopes is negative one. That is putting a gigantic cart in front of the horse. A much more elementary way of defining a right angle just be a right angle is an angle which is equal to its own supplement. So right angles are self-supplementary. It's that right angle is half of a flat angle, okay? And so once you have the idea of a right angle, we can talk about the idea of a complementary angle. Complementary angles are those pairs of angles whose union together forms a right angle. So for example, if we take say these right here, take that angle and then take that angle right here, these would be an example of complementary angles right there. So the union of the two angles gives you a, the union of two complementary angles always gives you a right angle. We can also define the notion of an acute angle. If you have, for example, a right angle like we did before, here's our right angle. A acute angle would be an angle that's bigger than a null angle, but smaller than a right angle. So such an example would be like an acute angle right here. It's smaller than a right angle. On the other hand, an obtuse angle is an angle that's gonna be bigger than a right angle but it's not a flat angle. So if we finish off and form a flat angle right here, an obtuse angle would be something like this. Like so, this angle in consideration. This would be an example of an obtuse angle. So some vocabulary that's important to introduce in this conversation about angles. And you might recall from our previous lecture about triangles, we often define triangles using angle terms. After all, triangle means three angles. The angles seem to play a big part of how we study triangles. There was the idea of an acute triangle. An acute triangle is one where all of the angles were acute with this meaning. An obtuse triangle was a triangle for which one of the angles is obtuse, the others three would necessarily have to be acute. A right triangle was an example of a triangle which had one right angle, the other two angles are necessarily acute. Triangles always will have at least two acute angles but the largest angle could be acute right or obtuse. So we say things like that. We don't have to worry about the, we don't have to worry about the third angle of a triangle being flat because if you had a triangle where one of the angles was flat, that actually wouldn't be a triangle. That would be something like this. Hey, we're a triangle. It'd be like a flat triangle which isn't technically a triangle but in some ways it kind of behaves like a triangle. What I mean is like if you take a triangle like this and you take this point, let's call it A, what if we were to move A closer and closer and closer and closer and eventually something like this keeping the scale the whole time. Eventually we might get to something like if you go through this progression might eventually get something like this. So a flat triangle is what we call in calculus the limit of a triangle. That is when we cross the triangle, we get this and that's what's often referred to as a degenerate triangle. The same thing with a null angle. A null angle is they don't really belong in triangles except for this degenerate case. So this situation like this where you have the three angles inside of a triangle in this flat triangle you're gonna have one flat angle in the middle but then the other ones are gonna be null angles. Again, these aren't technically considered triangles but sometimes when you take progressions of triangles you can get a flat one as you go towards infinity. That's gonna be a topic for another day. In conclusion of this video what I wanna make mentioned is that we took special attention never to talk about angle measures. We talk about just angles themselves. So as we define flat angles, acute angles, right angles we never talk about things like 180 degrees or 90 degrees or larger than 90 degrees. We didn't use those terms because although degree measure is a useful way of describing these things it's not actually necessary to describe that of angles but with that said in the very next video we'll introduce the idea of angle measure.