 Hi, I'm Zor. Welcome to Unisor Education. Today's topic will be angles in geometry. Angles in the planning metric, which means angles on the plane, that's what we are considering. All right. First of all, let's talk about the definition. Everybody knows basically how angles look like, but let's just start from the very fundamentals, the definition. We all know what is a straight line and what is a ray, which is actually a half line. It starts with a certain point and then goes to infinity. Now, two rays which share the same beginning form and angle. Is it sufficient to define this particular angle? Well, actually not. For obvious reasons, there are two different angles just here. One is from this to this and another from this to this. Well, is that it? Now, there are actually four angles defined here, because we can always go in an opposite direction. As you see, just a picture of two different rays doesn't really define an angle. What we have to know is what's the first ray from which we start, and what's the second ray where we finish. Usually, if you have an angle and you are saying something like a, o, b, what it implies actually is that you start from the a, which is o, a, ray, and then you go to the ray, o, b. So from ray, o, a, you go to o, b, because the letter a is first. All right. Now, what it means is that we should really wipe out this particular one and this particular one. So we start either this direction or that direction, but we start from the ray, o, a, and go to o, b. Now, which direction we go? Okay, here we need certain convention. And the convention is that the positive direction of movement is counterclockwise. Just by definition. There is nothing to basically talk about this. It's the definition of the positive direction of the angle. So if we are talking about this, this angle is this one, not this one. Because this one goes counterclockwise. This one goes clockwise. So whenever we are dealing with notation of the angle, we have the first ray in the beginning, the second ray at the end, and the direction from the first to the second is always counterclockwise. Okay, define that. Now there is an interesting thing. We have to measure these angles in some way. Now the first traditional, if you wish, measurements of the angle is the following. Basically, you can consider the angle from one ray to another, which coincides with itself, this angle, a, o, b, as 0. And the unit of measurement is degree, which is written as degree of temperature, basically. So the angle with coinciding sides is basically, by definition, a zero degree. And if the same looking angle is basically the result of a full circle from here back to here. So just imagine that this particular oe is moving all the way out here. And finally coincides with oe, again coincides. That's basically the same angle from the visual perspective. It looks as two rays coinciding with each other. But the whole movement around the whole circle is actually, by definition, defined as an angle of 360 degrees. Now, what does it mean? Well, it means that something like half of this angle, if you go from here to here, so you go from oe to oe, which is stretched on the same line, which is basically half of the full circle. Well, this particular angle is equal to half of this. And obviously quarter, which is perpendicular with the 90 degrees, etc. So this is how we measure the angle. And at the same time, you see that if you add 360 to an angle, it will be visually exactly the same thing. So angles, which are measured by the difference in 360 degrees, are not distinguishable among themselves. So what you can say is that angle in, let's say, 1 to 7 degrees is equal to, or congruent, or whatever the terminology used, 300 and 87 degrees, and in turn equal to the angle of 360 twice is 7.47 degrees. All right, so these angles are exactly the same from the visual perspective. Well, there is nothing wrong about this. That's the fact there are periodic functions, which have exactly the same value for different values of arguments. So in this case, argument is the number of degrees, basically, if you wish. But the physical angle as you see it on the plane is the value of this function, and they're all the same. All right, so we finished with this particular type of measurement. Now let's talk about arithmetic. In the previous lecture about lines and segments, I was talking about segment arithmetic, that we can add them, we can multiply by natural numbers, we can divide by natural numbers, or we can multiply by rational and even irrational numbers. So exactly the same thing exists with angles. What does it mean to add two different angles together? Okay, let's say you have an angle A or B and you have an angle C or G. Now, what does it mean to add them together? Well, obviously you have to have an angle and some ray from the same vertex O, called it G, in such a way that angle DOG is congruent to angle COD, and angle GOF is congruent to angle AOB. That's what it means to add two angles together. Well, obviously if angles are really very large, you might have an interesting picture of this. Well, consider the angle AOB is up tos and angle COD is even bigger than that. So you have one angle, which is greater than 90 degree, then another angle, which is even greater than 270 degrees. Now, what happens if you add them together? Well, let's consider the movement. So first, we draw the beginning grid. Then we have to add the first angle, so that would be something like this. Then we have to add another angle, like this. So it will be something like this. That would be our... So the resulting angle will be this one. That's how it looks. So this particular resulting angle is smaller than this one, and definitely smaller than this one. So it looks like you add two angles and you get the result, which is smaller than the component. Well, with angles it happens. And it happens only because 360 degrees, the full circle, is some kind of a limit. So if you go over that limit, well, you basically count only what goes over 360 or 720 or whatever it is, whenever you are representing in the visual sense. So you can say that this particular angle is actually the result of this movement and this movement. But one full circle actually should be completely disregarded, because from the visual perspective, angles which are different by a full circle don't really look any different. So that's why we have such a peculiar situation with these two angles. Let me just take care of these features. It looks kind of more interesting. So the sum of these two angles is equal to this angle. And well, that's how it is, basically. All right, fine. So we are talking about addition. Now, obviously, we can talk about multiplication by integer numbers. And obviously, it means that if you have an angle, let's say AOD, and you want to multiply it by some integer number, let's say, by 2, what it means, it actually means addition. It's AOD plus AOD twice. If this is any number, it will be as many components in this sum. So it works actually the same way as with segments. So you can add two different angles. You can multiply them by integer number. Now, you can do the reverse operations, subtractions, and division. And obviously, as far as subtraction is concerned, if you have two angles, this one and this one, this is AOD and this is COD. So how to subtract? Well, again, start from the gray. Have the first angle congruent to AOD, that's F. And now, instead of adding this angle into this direction, into positive direction, which is counterclockwise, you do clockwise into this direction. So it will be this angle. So this one is equal to this one. But this is positive, and this is the negative direction. So first, you move into positive direction by the value of the first angle. And then you move to the negative direction by absolute value of the same angle. And whatever is left here, that's the result of a difference. Why? Because if you add them back together, this is G. So EOG is basically the difference between angle A or B and COG. Why? Because if you will add this to this angle EOG plus angle COD, EOG plus COD, which is congruent to JOF, that would give you the EOF, which is congruent to our original angle AOD. So that's how you verify that this is actually the difference. Subtraction is verified by addition. Now, same thing with division. If you have an angle and you divide it by a certain natural number, what is angle A or B divided by, let's say, 3? It means you have to find such an angle COD. So this will be COD in such a way that if you will add to itself multiply by 3, which is the reverse of the division, three times you will get an angle congruent to our angle AOD. Now, let me repeat something which I have said when I was explaining the arithmetic of segments. Defining sum of two angles or difference between them or result of multiplication or division, defining is one thing. But every definition must be correct in some way or another. Now, what's the correct definition of, let's say, sum of two different angles? Well, we have to prove that this new angle which constitutes the sum of these two angles originally given to us really exists. And no matter how you do it, you will have the same and unique result of this operation of addition to different angles. So operation must be defined in such a way that the result exists and unique. Now, my question is, let's talk about division, for instance. Now, I was saying that I have to find an angle which is, if multiplied by 3, will give original. Now, does it exist or not? Well, I have to prove it. And secondly, is it unique? I mean, maybe there are two different angles which if multiplied by itself three times, I mean, if added to itself three times, multiplied by natural number 3, would give original number. What if there are two different angles which means there are two different results of the operation of division? Both statements must be proven, existence and uniqueness of the operation. Otherwise, operation is fully defined. Well, not defined, let's just be honest about it. Well, and it's not easy, by the way, to define these most important for any definition properties, existence of the result of the operation and its uniqueness. But it can be done. The definition itself is quite obvious, but all these proofs of existence and uniqueness are not really easy. They kind of go beyond the scope which I would like to address. But in any way, you should understand that this is very important. Again, let me repeat, uniqueness and existence are very important for any kind of a definition. Otherwise, you will be defining something which does not exist or something which exists in different incarnations, which is basically not a good thing in mathematics. All right, fine. So we've done that. We've done the operations. And one more thing about how to measure the angle. So we were just talking about measurements in degrees when the full circle is basically defined as the angle of 360 degrees, and everything which is part of this full circle basically defined through the operation of multiplication or division or whatever else. So if you can say that if you have angle A, O, B, and you can say that this is basically an angle of how should I specify it? Let's say I put A, O, A, which means a full circle, full circle times some number R. Well, just as an example, if it's something like this, it looks like, I don't know, 60 degrees or whatever. It means that 360 is multiplied by 1, 6, whatever the number is. Well, then the measurement is 360 times R, this angle. Because the measurement of the full circle is 360 degrees. So basically, you can say that you have a unit of measurement, which is a very small angle, which if multiplied by 360 will give the full circle, will give the full angle. And this is the unit of measurement. And you can either start with this small 1 degree angle and then express our angle through this. Or you can start with a full circle. Maybe it's easier. It doesn't really matter. And multiply it by some multiplier. So either you multiply this unit of measurement by some number to get the number of degrees. Or you multiply this, which is expressed as 360 degrees by a different number. In any way, whatever the result is, you have to get our angle A, O, B. And the measurement would be calculated as either if you start in the full circle, then it's 360 times the multiplier. So in order to start with this small angle, let's call it X or Y. So if you start with this small angle, X or Y times some measurement, some number S, then it will be 1 times S. So in any case, that's the measurement in degrees. Are there other measurements of the angles? Well, if you remember from the previous lecture about the segments, I mentioned that there are so many different measurements of the length of the segment, it's unbelievable. Well, with angles it's much simpler. There are only two different units of measurements which people usually use. One is degrees. And another is based on an interesting property. If you consider a circle and the correspondence between the radius of the circle, and the lengths of the circumference of the circle, there is an old fact, basically, that circumference is equal to 2 pi r. r is a radius. Pi is an irrational number which is approximated as 3.14. And so the lengths of the circumference is actually proportionate to the radius. Now, here is an interesting thing. What if we will take an angle which has radius equal to the length of this arch? Well, if you increase the size of the circle, then what's interesting is that this length is increasing in exactly the same proportion as the radius. So if you take a smaller one, the radius is, let's say, twice as small. And this length will be also twice as small. So basically, if you take this type of an angle which cuts from the circumference, an arch which has a length equal to the radius, then this angle will be exactly the same regardless of the circle. So this type of a definition seems to be much more convenient in certain aspects of angle arithmetic, angle calculation, and equations, et cetera, et cetera. And we can actually measure any angle using this particular angle as a unit of measurement. And it's called this unit of measurement is called one region, one region. So this angle, which cuts from the circle, an arch equal in length to the radius is actually independent on the circle. And it can be therefore used as a unit of measurement to call it one region. Now, if this is one region, then how many regions are in the full circle? Well, let's just think about if the full circle is equal to 2 pi r circumference. And this particular length is equal to r. Then if you divide one into another, you will get 2 pi. And this is basically the number of regions in the full circle. So whatever is 360 degrees is equal to 2 pi regions. That's what's very important now. It's just a different unit of measurement. And we can measure angles in this unit as well as in some other units. Now, although it seems a little strange to have an irrational number as the measurement of a full circle, it just takes my word for it that in many different aspects of mathematics, this seems to be as more natural unit of measurement. And it's basically much wider used in mathematics for different reasons, which we will not discuss right now. But that's actually true statement. Now, from this, from proportionality, you can say that 180 degrees is actually pi regions. And let's say 60 degrees is pi divided by 3 regions, et cetera. So two different measurements, degrees and regions, are the measurements which are used in mathematics. In common life, degrees are more prevalent. But in the mathematics, especially in high mathematics, which people are learning in like calculus, et cetera, then this measurement, the region measurements of the angles, are more prevalent because it's more convenient in many different cases. Well, that's it for the angles. I would like to remind you that this lecture and notes for it can be found at Unisor.com website, which actually presents a course of school level mathematics more or less logically arranged. And it's also very helpful to those parents and supervisors who would like to control the educational process of their children and students. Because there is something which they can use as tests and exams, scores, et cetera. There is some kind of a relationship between the supervisory activity of the parents or supervisors and educational process, which they can control. So I strongly recommend to go to the website and basically investigate how to use it and use it. Thanks very much.