 Hi, I'm Zor. Welcome to InDesert Education. Continuing course of advanced mathematics for teenagers, we are talking right now about solid geometry, and in particular about an angle which can be formed by two straight lines in the three-dimensional space. Now, first of all, let me tell you that two lines, which are in the same plane, form the angle which we know basically everything about it, right? So, our purpose right now is to transfer the definition which we know about two-dimensional case into a 3D. Well, there is one very important difference. In two-dimensional space, two lines are either parallel, in which case we can say that they form an angle of zero, basically degrees or regions, whatever. Or they intersect, and then there is an angle. Now, intersection basically means there is a point which belongs to both lines. Well, in three-dimensional space, that might not be necessarily the case. If you consider these two lines, they do not have any common points, so they don't actually intersect. Now, how can we measure an angle between two lines which do not have any common points, no intersection? Okay, and here is the procedure which we are going to use to transform this definition from the two-dimensional space into three-dimensional. First of all, let's consider an angle, well, for simplicity, let's consider the angle between rays instead of two lines. So, you have one ray and you have another ray. And let's consider that these two rays have common origin or vertex. Now, in two-dimensional case we know what exactly is the angle between these two rays. And in three-dimensional case, first of all, let's consider the situation when they also have common origin. So, we have some situations like this. In three-dimensional space, one and another, and they have common origin. Not like this. So, first they have common vertex of these two rays. Now, from the previous material, we know that if you have two rays in three-dimensional space which share the common origin, there is one and only one plane which they belong to. Well, incidentally, how to basically prove it? Well, we have to go to the axioms of solid geometry. It can be proven very easily. Let's take two other points except origin, one and one ray and another and another. There is an axiom that three points define one and only one plane which contains these three points. That's number one. So, we have at least one plane. And then the second axiom is that if two points of a line, straight line, belong to a plane, then the entire line belongs. Same thing here. Two points belong to a plane. So, the entire line between them belongs to the plane. That's how we prove that there is one and only one plane which contains these two rays with the common vertex. Okay, so now, even if we are talking about three-dimensional space, in theory these two rays belong to one and only one plane, some plane. Okay, now let's just say that three-dimensional concept of an angle between these two rays with the common origin in a three-dimensional space is actually exactly the same as their angle in the plane, one and only plane, which contains these two rays. I mean, this is something absolutely natural and trivial. We have two rays, so in three-dimensional space the angle between them is basically the same as the angle within the plane they belong to. Okay, now our next problem is to define an angle between two rays which do not have a common origin, something like this. Okay? But this is actually very simple now because, for instance, we have this situation. Forget about this plane, there is no plane which contains both of them. So, this is one and this is another. Now, let's pick any point in space and draw one ray which is parallel to this one and similarly directed and another which is parallel to this one and similarly directed. Can we do that? Can we draw a line parallel to another line through a point given outside of that line? Yes, we did consider that situation already before. Very quick return back, flash back. You have a point and you have a line, so there is a plane, one and only plane which contains this line and this point and within that plane we draw a parallel line, we know how. Same thing here. So, now what we are saying is that by definition the angle between these two rays in three-dimensional space which do not have the same origin is an angle between these two rays where I chose completely randomly an origin and draw a line parallel to this one and draw a line parallel to this one and similarly directed. So, that's the definition. Now, what's important about this definition is that, well, it has certain random character. We choose randomly this point. Now, my question is what if I will choose a different point in space and do exactly the same? One ray would be parallel to this one and one ray would be parallel to this one. Would I get a different angle? I mean, in theory these two are completely different constructions. Okay, now we have to prove basically that the angle will be the same. But this is actually very easy. Why? Because since this is parallel to this and this is parallel to this, these two are parallel among themselves. We have already proven that in one of the theorems before. So, these are parallel. Similarly, these are parallel because each one of them separately is parallel to this. So, now we have two different angles in three-dimensional space with correspondingly parallel sides. And again, we have already proven before that these two angles are exactly the same. They are congruent. So, that proves that our definition makes sense. Because if there is no such equality between these two, then the definition doesn't make any sense, obviously. So, because the concept of an angle would not be defined properly. It could be one and then another. But in this case, considering there is this theorem about congruence of these two angles, that means that the angle is completely defined. Now, why did I choose to work with rays rather than lines? Well, it's just a little complication which we have maybe a nuance if you wish. It's very easy to deal with. Let's go back to the plane. Now, if on the plane you have two rays which form some angle, well, how many angles they form? Four, to be exact. One angle. Another angle. The third angle. And the fourth angle. So, let's say this is 30 degrees, right? Then this would be 360 minus 30, which is whatever, 330 degrees. And this is the positive direction. This is the negative direction. You see, it's really a lot of different angles. Now, if you take two lines, it would be even more difficult to say what kind of an angle between them. Because you have one, two, three, four angles, well, granted, two of them are the same. So, at least two angles you have. And each one of them can be counted in a positive or a negative direction. So, there is certain fluidity, if you wish, in this particular case. Now, what do we do in three-dimensional space? We want to simplify our job. Now, to simplify our job is, first of all, forget about direction. If you are in a plane, you can actually look from the top and you see counter-clockwise movement. If you are in a space in this situation, there is no such thing as counter-clockwise or clockwise. So, there is no direction, so to speak. So, forget about the sign of the angle. And now, considering this and this, we usually choose the smallest one as the angle between two rays. So, whenever we talk about two rays or even two lines in three-dimensional space, we forget about the sign of the angle and we choose the smaller one. So, in this case, it's this. Now, if it's in space, something like this, again, we have a point. One is parallel. Another is parallel to this. So, this would be the angle between these two lines. So, that's how we define precisely the angle between two lines in space, which are not intersecting each other, which are skewed. So, finished about definition. We have defined an angle and we prove that the definition makes sense. Okay. Now, what's next? Next is to prove a couple of theorems. Okay. We have three theorems, which I'm going to prove here. Very simple. One is if you have a plane and a perpendicular to this plane. So, a plane is gamma, perpendicular is A. And any other line on the plane B. So, what I'm staging right now is that the lines A and B, A is perpendicular to the plane gamma and B belongs to gamma. They are always perpendicular to each other. The angle between these two is 90 degrees, the right angle. Now, by definition of the perpendicular line, we know that any line which goes through this point on the plane gamma is perpendicular to A. That's called B prime. That we know. That's actually the definition of the perpendicular. Perpendicular to the plane is perpendicular to any line which is in that plane and goes through the intersection to the base. Now, but look, this is parallel to this one, right? So, an angle between this line and this line can be formed by our process of bringing together lines from this point. It's called A. Well, we already have this point on this line. So, all we have to do is draw a parallel line to this one, right? And the angle between A and B is, by definition, an angle between A and B prime. And A and B prime form the right angle because A is perpendicular to the plane. So, that's why the angle between A and B, the angle in three-dimensional sense, even if they do not intercept each other, they're skewed, still is the right angle. Very simple theorem. And only one additional construction I need is to draw a line parallel to line B through the base of the perpendicular A. Simple. Next. Okay, what's next? Okay, very similarly, we have some kind of a plane and perpendicular. Now, what I was talking before that any line on the surface of the plane gamma is perpendicular to A. Now, we're talking in this particular theorem about any line which is parallel to plane gamma. And the statement of the theorem is that any line parallel to the gamma would be perpendicular to line A which is perpendicular to gamma. Again, it's a very easy theorem to prove. How? Well, let's say this is point A. Now, we can always draw a plane through A and B. Now, again, B is parallel to gamma. And the intersection of this plane with gamma would be something like this. That would be my B prime. So B prime is intersection of gamma and plane which goes through line B and point A. Now, there was a theorem again before that if you have a line parallel to a plane and you have a plane which goes through that line intersecting this plane, then the line of intersection would be parallel. Okay, fine. So this is line of intersection. It's parallel to B. But now, this line of intersection is intersecting perpendicular A right at the base which means it's supposed to be perpendicular. A and B prime are perpendicular. That's why A and B are perpendicular. Also very easy. Next. Now, we have... Well, it's kind of a reverse theorem. We have two lines which are perpendicular to each other. So this is our plane. This is our A. And I know that B perpendicular to A. Now, my theorem is if B is perpendicular to A, not necessarily intersecting. Somewhere, let's say A is within this plane of this board and B is above and beyond somewhere in the space, not on the board. Okay. Now, in this case, B would be parallel to gamma. So A is perpendicular to B. A is perpendicular to gamma. Now, even if A and B is empty, A intersecting with B is empty. I still have that B is parallel to gamma. Okay. Now, here is how we can prove it. Well, let's do it in an easier case. What if B does intersect? If B does intersect A, what I will do is the following. Through A and B, I can draw a plane and it will intersect this plane. It will intersect at B prime. Now, this angle is right because A is perpendicular to entire plane gamma. Now, this angle is right by assumption which I just made. Now, A and B are perpendicular and I made an assumption that they actually have an intersection. So, these are two perpendicular to the same line A and they are aligned within the same plane, obviously. So, they are parallel to each other. It's a plane geometry, two-dimensional case. In this plane, we have two perpendicular to a line, so they are parallel to each other. And since they are parallel to each other, B and B prime. I can say that the angle, I can refer it to a theorem which states that if line one line outside of the plane is parallel to a line on the plane, then this line is parallel to an entire plane. Again, that was one of the theorems before when I was talking about parallelism between lines and planes. So, if line is parallel to one line on the plane, then it's parallel to an entire plane. So, in this easier case, we have proven that this theorem is true. Now, let's consider a little bit more complex case. What if they do not intersect? So, B is somewhere behind, let's say. Now, what I will do is, well, very simple thing. I can just pick any point A on the line A and draw a parallel to line B. Well, not exactly parallel, this is parallel. This would be my B prime. And then, I will draw a plane through A and B prime to get the B second. And basically continuing the theorem after that. So, this angle, obviously, is perpendicular because I have just already proven that. And that's why angle between A and B. Because angle between A and B is exactly the same by definition of the angle in the three-dimensional space. It's exactly the same as angle between A and B prime. So, B prime is also perpendicular to A as well as B. So, we basically refer to a smaller, a simpler version of this theorem. We have one line which is perpendicular. And we have proven that this line is parallel to entire plane. But if this is parallel to this and this is parallel to this, then B is parallel to B second, which therefore proves that B is parallel to an entire plane gamma. So, these are very easy three theorems. Well, I hope you don't have an impression that every theorem which we will be talking about is simple. I mean, maybe we will talk about simple theorems, but they are much more complex. Well, this three-dimensional case, it's really complex because it's always difficult to have this image of how these components intersect with each other. Like, for instance, in this case, I'm just saying that line B is not intersecting A. It's either behind the plane which is this board or above it. But to imagine it is not probably difficult. I mean, it's not probably easy. So, that's why I would prefer to start with these really simple problems just to basically develop your space vision. That's what's very important in solid geometry, space vision. And by the way, this space vision would probably be very useful for you anywhere wherever you go. I mean, if you are an architect or if you are, you know, building some machine or constructing something, I mean, it's always very useful to have this three-dimensional vision. All right, that's it for today. I wish you registered to Unizor.com website because in this case you will be able to have the functionality of like a real school which means you can enroll in certain subject like solid geometry, for instance, and you can take the exams. There is a supervision role in this particular case. Some supervisor, your parent, maybe your teacher can look at the results of your exams and you can take exams as many times as you want, basically. Just basically, you know, viewing whether your results are improving or not. All right, so that's it for today. Thank you very much and good luck.