 Hi, I'm Zor. Welcome to Unizor education. I continue the course of advanced mathematics for teenagers presented on unizor.com and right now we are talking about solid geometry. This is actually a second introductory lecture. These introductory lectures are supposed to basically introduce you to main objects the solid geometry is dealing with like planes, lines, spheres, etc. So, this particular lecture is about angles. Now, we do know about angles on the plane between different lines. Now, that's actually a very simple thing. And if you imagine, let's say, a ray and another ray which originates from the same point, you can always measure the angle between these two rays as basically a measurement of rotation of one ray to rotate until it coincides with another ray. Now, obviously, you can assign a measurement of the full circle until it coincides with itself as, let's say, 360 degrees or 2 pi radians. And then you can establish actually the positive and negative direction like counterclockwise is traditionally is positive and clockwise is traditionally negative directions. So, we can all do this with rays and lines on the plane. By the way, as far as lines are concerned, situation is just slightly more complex because you have two different angles. If you have two lines, how can you say what is the angle between these lines? Well, there are many angles, this angle, this angle, this angle, this angle. And then there are two directions. So, from here to here or from there to here. I mean, it's different directions, different angles. But anyway, it's all very much defined and very well understood. Now, switching to the solid geometry in space, we are talking about angles between the planes. Now, before doing that, I would like to actually remind that we have established, actually it was a previous lecture, we have established certain very important axioms, not all the spectrum of all the axioms, but just specifically axioms related to planes in solid geometry. And let me just refresh your memory about these axioms. So, if two points A and B belong to the same plane, let's call it alpha, so A and B belong to the plane alpha. Like in this case, plane alpha is my white board and these are two points. Then, the line which connects these two points, let's call it line B, and it's connected points A and B, that's how I'm using these symbols. So, the entire line belongs to the same plane, belongs to the plane alpha. So, these two points are on the surface of this white board, and then therefore the line which connects them also belongs to the same board. That's an axiom, which we call A1. Axiom A2, if two different planes are intersecting at one particular line, and let me try to draw it in some way. So, this is one plane, and this is another plane. So, what I have to do, I have to do this as an invisible line. Well, okay, something like this. So, if they are intersecting at one particular point A, then they actually intersect along the straight line which goes through the point A. So, if point A belongs to the intersection of two planes, alpha and beta, so I'm using the set theory symbol of intersection, then, I didn't have this sign then, line which, let me use square brackets, then the line which is passing through the point A also belongs to intersection. So, if they intersect at a particular point, they intersect at the line which is passing through this point. So, this is axiom A2 and A3. If you have three lines, sorry, three points in space A, B and C, which are not lying on the same line, then there is one and only one plane which passes through these three points, which contains these three points, one and only one. So, three points not lying on the same line define the plane. And we actually had some very simple micro theorems or nano theorems, which are very simple, like for instance, if you have a line and a point outside of this line, then you can always draw a plane which is passing through this line and the point. I mean, there are some others. But, okay, I just wanted to remind you these three axioms as an introduction to this concept of an angle between two planes. So, right now we are talking about two planes and we would like to somehow measure the angle between these two planes. Now, let me return back to the concept of rotation. The same concept I was using as a measurement or illustration of the angle between two lines on the plane. So, two lines on the plane which are intersecting, you can just rotate one until it overlaps the other. We will do exactly the same with planes. And the axioms which I was just talking about will help us. So, let's consider that we have two planes which do intersect, which means there is at least one common point. Now, we know from the axiom A2 that if there is a common point between two planes, there is a common straight line. So, let me draw again this particular and I will do it something like this. And this will be invisible, right? So, one plane goes this way and another is basically the plane which coincides with my whiteboard. Now, this is an intersection. It's a straight line and what we can do actually, we can try to imagine a rotation of one plane until it coincides with another plane. So, it's like a book. You have the book, so you open the book and that's basically a rotation of the one page relative to another page. Now, you can obviously measure this rotation. You can always say that, okay, the entire circle, if you have, I think it's a little bit easier if you deal with half planes, right? I don't really draw the continuation of these planes beyond the line where they intersect. So, you have two half planes, right? And the question is how to measure the angle. Well, exactly the same way as we did with rays on the plane. With rays, we had the full circle until it coincides with itself as 360 degrees or two pi radians, right? We'll do exactly the same here. So, we'll take this particular plane. Sorry. If you can rotate it around this particular line as around the axis completely, the full circle until it coincides with itself, that will be an angle of 360 degrees. So, that's how you measure, that's how you introduce the unit of measurement. So, basically, the angle between this plane, this plane and this plane can be measured in the same units, like what part of the full circle it contains. So, this is kind of a definition, if you wish, of what is the angle between these two planes. However, it's always interesting to compare it with angles between different lines on these planes. Now, I will tell you something which I have no intention to prove right now. That would be in the proper time in the proper lecture, just jumping a little bit forward. What's interesting is, if you will draw another plane, so let's say this is plane alpha, this is plane beta, and I would like to draw another plane which is, I think something like this, something like this. We will call it gamma. And what I'm saying is, it intersects this line perpendicularly, and I did not define the perpendicular between the line and the plane, but basically intuitively, you understand, that if this is my plane, this is the perpendicular, not this one, not this one, but this one. So, this perpendicular to this particular line of intersection between alpha and beta plane is also intersecting alpha and beta. So, this would be the line where it intersects alpha, and this would be the line where it intersects beta. Now, these two lines, call it A and B, they align in the same plane gamma. So, what I'm saying is that measurements of this angle between line A and line B, which are intersections of gamma with alpha and beta correspondingly. This angle in measurement units of the angles on the plane is exactly the same as the measurement between these two planes measured in the units of the angles in space between the planes. I'm not going to prove it, I'm just saying that this is actually something which would help you to feel what is exactly the angle between the two planes. And I think it's quite understandable if you will take a book again. Open it up, and then if you imagine from the center line, if you can draw the perpendicular to this line on the one page and on another page, the angle between these lines would be very much the same as an angle between these planes. So, that's how you basically understand that there is definitely a connection between the angles between the planes in space, in three-dimensional space, and lines on the plane. Well, that probably is, as I was saying, a little bit jumping forward. And I will definitely explain all the details in the corresponding lecture. The purpose of this lecture was just to introduce you to the concept of angles between the two planes. Now, obviously, I didn't mention it, but if you consider two planes which do not have any common points, how would you call these planes? Well, parallel. Same as two lines on the plane which do not have any common points are called parallel lines. So, that's something which is again related to the angles. The angles, you can say, they're making the angles of zero degrees of radians or whatever. Let's just put all these little issues aside. The purpose of this lecture is introducing you to a concept of an angle. That's it. No properties, no theorems, just the introduction of the concept. Okay, that's it. Thank you very much and good luck.