 Hi, I'm Zor. Welcome to Unizor education. I would like to continue talking about planes and lines in the course of Solid Geometry. I suggest you to watch this lecture from the website Unizor.com not from the YouTube channel or anywhere else because the website contains very detailed notes for each lecture. So it would be better if you just read the notes before or after the lecture. Now the previous lecture was about two lines and a plane. And I was talking about when they are perpendicular, parallel, etc. Now this lecture is basically like a mirror image of the previous one because now I have two planes and a line. And again it will be very similar theorems. The same three theorems as before in the previous lecture but instead of planes there will be lines and instead of lines there will be planes. So, ok, here they are. By the way, in the previous lecture the first theorem was if you have two parallel lines and one of them is perpendicular to a plane then another one will also be perpendicular to a plane. Now here the first theorem is very similar but as I said planes are lines and lines are planes. So let's assume that you have one line and you have two planes parallel to each other. Something like this. So this is one line and two planes. Gamma and delta. So gamma and delta are parallel planes which means they do not have any common points, no intersection. And also I know that the line is perpendicular to one of them. I have to prove that it's perpendicular to another one as well. Ok, so this line intersects gamma, plane gamma at point p and let's say it intersects the line delta at point q. Ok, the question is does point q exist? So if my line intersects one of two parallel planes it must intersect another one. Well, how can we address this issue? Let's just think about it. If it's not then if there is no q, if there is no intersection then this line is parallel. How can that be? Well, let's just leave it alone. Let's just assume that there is such a point q. Actually, I think it would be a very good exercise if you think about how to prove that the intersection actually exists. So let's assume it exists. So what we will do next is we will create a plane through the line A and the plane, since it completely contains line A it must contain points p and q on it. So it will intersect. So this would be my plane and it will intersect this and this. So this plane, let's call it I don't know how alpha. It intersects q, it intersects gamma at some p, p star and it intersects delta at q, q prime, p, p prime and q, q prime. Now, these lines are supposed to be parallel because we already discussed this theorem when we were talking about parallel planes. So if you have two parallel planes and these parallel planes are intersected by the third plane then the lines of intersection must be parallel to each other. Well, actually this is a simple thing to prove. I mean, if you think about this these lines are lying in the same plane, alpha. They must not have common points obviously because if they do then the planes would intersect and they are supposedly parallel. Alright, so fine. So p, p prime and q, q prime are parallel to each other. p, p prime is parallel to q, q prime. But p, p prime is perpendicular to the line a because a is perpendicular to an entire plane. Now, if you have a line and you have one perpendicular to the line and another line which is parallel to the first one and everything is going on within the plane alpha. So this is the plane geometry. So obviously if you have a line, one is perpendicular and another line which is parallel to this one. It also is perpendicular from the course of the plane geometry. So that means that q, q prime is perpendicular to our line a. Now, we will do exactly the same with another plane. Let's try another plane and we will have another intersection. p, p prime and q, q second. Absolutely the same logic. It's just a different plane. Let's call it beta for instance. And from this we see that q, q second, double prime, whatever is also perpendicular to a. So consequently we have a line which is perpendicular to two lines on the plane and that is a sufficient condition for the line to be perpendicular to the entire plane as we have proven before. Okay. All right. Okay. So let's go to the next. Yes, we still have this hanging issue with the intersection of the line a with the plane delta. By the way, if anybody comes up with some kind of a very nice and concise proof of this, send it to me and I'll just put it on the website. Meanwhile, I'll think about this myself and I'll put it into notes for this lecture. If it's already not there, I don't remember actually. Anyway, number two. Again it's in a way similar to the previous lecture, the theorem number two, but instead of lines we have planes and instead of planes we have lines. Now in this case I have one line and I have two planes which are perpendicular to the same line. I have to prove that the planes are parallel. Okay. So how can we prove that the planes are parallel? Well, in this case we will do the following. Let's assume they're not parallel, which means there is common point. There is an intersection and there is a common point. So let's say these planes somehow are intersecting somewhere. And there is a point somewhere where they intersect. Well, actually planes intersect along the entire line, but I don't really need the line, I need just one point of intersection. What I do next is, using this point of intersection M, I will draw a plane based on the line and the point of intersection M. Now this plane will intersect gamma somehow and it will intersect delta somehow. So the plane which includes line A and point M should have certain intersections with both gamma and delta. Now, since I was saying from the very beginning that A is perpendicular to gamma and A is perpendicular to delta, it means that A is perpendicular to Pm and A is perpendicular to Qm. So within this plane, which is based on the line in M, I have two perpendicular to the same line A from the point M, which in the plane geometry is impossible, right? We have a line, we have a point, we can have only one perpendicular, we don't have two different perpendicular. Which means that our assumption about the intersection of these two planes is just wrong, they do not intersect, which means they are parallel. Okay, and the third theorem is, if you have a line and a point outside, which is not on this line, and you know that there is a plane, this point belongs to, which is perpendicular to A. Now let's say we have a completely different plane, which also originates, well, it contains the point M, different from the gamma. So there is another plane, whatever the plane actually is, something like this, called delta. So delta is also a plane which contains point M. So my point is that if this is a different plane from the gamma, then this plane cannot be parallel to gamma, and it cannot be perpendicular to the line. So again, gamma is containing the point M and is perpendicular to A. Delta is a different plane, which also contains point M. So this new plane delta cannot be parallel to gamma and cannot be perpendicular to A. Well, it's actually a trivial theorem, quite frankly. Well, number one, it's obviously not parallel to gamma because gamma and delta have a common point, right? They're intercept, so nothing to talk about, there is no parallelism at all. How about perpendicularity? Well, let's just think about if delta is perpendicular to A, and we know that if two planes, two different planes are perpendicular to the same line, they must be parallel, that we have just proven that there is no parallelism among these two planes. So it cannot be perpendicular to A. That's basically it. Now, what's important right now is not that these are, well, difficult or anything like this theorem. They're not difficult at all. Actually, they're trivial and everything follows from definitions and probably one or two theorems which we have already proven. What is, however, important from educational standpoint is for you to basically repeat these proofs and preferably not even to yourself but on the paper. If you write these proofs, that will be even much better. You will learn it better, actually, because when you're writing something and you're trying to prove your point, it's actually a tremendous help in development of your analytical thinking, your logic, etc. So I strongly recommend to do just that. You can listen to lecture, you can read the notes, but then put everything aside, have a piece of paper and try to prove these simple theorems just by yourself and that would be the great help for you. That's it. Thanks very much and good luck.