 Hi, I'm Zor. Welcome to a new Zor education. This is the first lecture which basically talks about elements of solid geometry. Before we had some introductory lectures, which I was I was just explaining what kind of objects exist in solid geometry, what we will learn and this lecture is the beginning of a relatively detailed studying of different elements. Now today, we will talk about parallel lines in three-dimensional space. Now this lecture is obviously of the course of advanced mathematics for teenagers presented in Unizor.com. I do suggest you to to watch this lecture from this website because the website contains very nice notes for the lecture, so you don't really have to write anything. It's already written, can be used as a textbook, of course. I would also like to encourage students to register on the website with their supervisor, teacher, parents, whatever, who can basically enroll them in different subjects, which are studied at this particular course. That would enable them to take exams. That's what's very important, I believe. So if you register student, you can take exams and it's free, by the way. All right, so let's talk about parallel lines. So line, parallel line and we are talking about three-dimensional space. Okay, first of all the definition. Which lines are called parallel in three-dimensional space? Well, you remember that in the plane geometry, geometry on the plane, the parallel lines are those which do not have any common points, no intersections. Now, this is not a sufficient definition for a three-dimensional space and I will later on address this issue right now again back to the definition. So the definition of the parallel lines in a three-dimensional space is, well, the first is the same as it was in a two-dimensional space. They don't have any common points, so don't intersect. But the second very important part of the definition is that these two lines, which are supposed to be parallel, they're called parallel, they are supposed to be in the same plane. So there must exist a plane which contains both of these lines and they do not have any common points. Now, immediately from here follows that these two lines are parallel in a two-dimensional sense on that particular plane, but this is kind of trivial, obviously. All right, so basically we have given the definition of certain object parallel lines. Well, there are two very important considerations which we have to really study. Number one, do parallel lines exist and if they do, what's the procedure to construct them and is there any uniqueness in the constructing of parallel lines provided certain elements are given. And for this purpose we will resort to one particular task. If you have a line A and a point P in three-dimensional space, the question is, is there a line parallel to A which is passing through point P, no matter what A and P are given in the space? That's number one. Number two is is it a unique parallel line which is passing through P and parallel A? So both questions are answered positively. So exists and unique. So now we have to prove it. That's the best part of it. So let's talk about existence. So to prove that something exists, one of the way to prove it is just basically to construct it. So let's again assume we have a line and we have a point and now I'm going to construct the line parallel to A which is passing through P. How do I do it? Well, you should remember a few axioms which I have mentioned in one of the previous lectures. One of those axioms says that axioms, so we don't really prove anything, that if you have a line and a point outside of it, there is one and only one plane which contains this line and point P. So if line and a point outside of it are given, then we consider that we have a plane which contains them and there is only one such a plane, one and only one. So using this axiom, first what I'll do, I will create this plane. So now I have a plane where these two elements belong to. The line in this plane and the point, let's call it gamma. Okay, so we are using the axiom. So we don't really have to prove anything that plane exists by axiom. Now, now we have a plane. So now we can resort to our geometry on the plane. Now on the plane we know this particular task. So if you have a line and a point P outside of this line, we know how to build a line parallel to A which is passing through P. I'm not going to go through this procedure. If you're interested, you can go to the part of the course which is dedicated to plane geometry. So I assume that we know that there is a way how to do it. So there is one and only one line which is passing through P, let's call it B line, in that plane gamma which is parallel to A. Now, what's the properties of this line B? Well, since B and A are parallel in the plane gamma, it means that number one, they belong to the same plane and number two, they do not have common points and that's exactly what is the definition of the parallel lines in 3G dimensional space. We need belongingness to the same plane and non-intersection. That's exactly what we now have. B belongs to plane gamma and it has no intersection with A because these are parallel. If there was any intersection, it would be intersection in the gamma itself. So obviously it would be against the principle of the parallelism. So we have constructed a line B which is parallel to A and passing through point P. Yes, we did use auxiliary construction of the plane gamma. Fine. That's the way how to do it. Now, the second question is, the second exercise in logic, if you wish, is to prove that this particular line B is unique. Okay, let's consider we have another line, let's say C, which is also parallel to A and it's also passing through point P. Now, by definition of the parallel lines in 3D space, A and C must belong to the same plane. Right? Okay, fine. So for instance, they do belong to the same plane. Let's call it delta. So what can we say about two planes, gamma and delta? Well, number one, line A belongs to both of them. Right? Number two, point B belongs to all of them, to both of them. So it looks like both of line A and the point P outside of it belong to two different planes, gamma and delta, which is impossible according to the axioms which we have accepted from the very beginning. If you have a line and a point, there is one and only one plane which is passing through both of them, which means that gamma and delta are exactly the same plane. Alright? So if these are the same plane, let's call it gamma, you know, doesn't really matter how we call it, it means that B belongs to this gamma and C also belong to this gamma and obviously A. So it looks like in the same plane, gamma, we have one line A and two lines parallel to it, which passing through the same point P. Again, back to fifth postulate of Euclide on the geometry on the plane, that is impossible. We have to have only one, one and only one line. In Euclidean geometry, we have only one line which is parallel to a given line and passing through a given point. So again, back to the axioms, we resorted all these logical concepts back to the axioms and from these axioms follows very quickly that B and C are also the same and that proves the uniqueness of the parallel line which is passing through a point P outside of a given line A. So what's interesting is you remember this logical building, which we are building, starting from the axioms, then the first level of theorems, then the theorems which are based on these theorems and axioms, and then the theorems which are built on those theorems and theorems and theorems and axioms, etc. So we are building more and more. Right now we are in the very beginning of this construction. So these preliminary or very simple, if you wish, considerations about the parallelism, they are like the first floor above the axioms because the only logic which we should use is, okay, according to that axiom it's this way, according to that axiom it's that way. So basically we are just starting, constructing this building and these are the first and very elementary properties which we are talking about. In this case, properties of the parallel lines and we are basing our logic only on axioms. No other theorems actually are at our disposal. Well, basically that's it. It's a very short lecture just to introduce you to a concept of a parallel lines in space. Again, let me just finish it up with repeating what I said from the very beginning. Parallel lines in space must belong to the same plane and obviously they should not have common points. Now, what happens if we skip the first part? What if they don't want, if they don't belong to the same plane? Well, here is a typical situation. Now, we are in space in three dimensions. So a line can be like this and another line can be like this. You know, these two lines are hardly parallel. I mean, nobody would call them parallel. I would rather call them perpendicular quite frankly, but they don't have any common points. They don't belong to the same plane though, which means that the first part of this definition that the parallel lines should belong to the same plane is not true in this case. However, they are completely non-intersecting. So this is just a separate case of configuration between the different lines and there is a special name for it. These two lines, which are not parallel but still do not have common points, no intersection, they're called skew lines. So this is exactly the example of the skew lines. So skew lines are those that do not have any common points. Parallel lines are those which do not have common points and are aligned in the same plane. So that's the difference. Parallel skew. Well, that's it. Thank you very much and good luck.