 Hi, I'm Zor. Welcome to Unisor Education. Today's lecture will be about parallel lines, and this is an introduction to parallel lines, and basically all the time I will talk about parallel lines and transversal. Now, before going into the details of this, I'd like to spend a couple of minutes just discussing certain philosophical aspects of geometry or mathematics, if you wish, in general. It's all about proving something, basically. And when you prove something, you're bounding your proof in something which was proved earlier, and then something before that, etc. And that's how you go, basically, to statements which are supposed to be accepted as axioms, because there is nothing to prove them wrong. There is nothing before that. So, geometry is very obvious of this particular quality of the mathematics, because there are certain axioms or postulates, if you wish, which the whole thing is based on. And it's very important to do it right, to do, like, these are axioms, and these are theorems which can be proved based on these axioms. And those are theorems which are supposed to be based on the prior theorems and prior axioms. So, it goes only one way. We don't want to have any kind of logical loops, like A is because of B, but B is because of A. This is something which is completely no-no in mathematics. Now, things about parallel lines are very much related to axioms which were first put out by Euclid, and then made much more rigorous by Hilbert. In particular, what's very important in this particular topic is the fifth postulate of Euclid, which in its original form states something like this. If you have two lines and transversal, which crosses both of them, and if some of these two angles is less than 180 degrees, two right angles, right, B, 90 degrees, then the lines eventually will intersect somewhere. It's a pretty complex statement for an axiom. However, Euclid did not have any other way, basically, but to put it as an axiom because he could not prove it. And in theory, that's absolutely correct way to do this type of things, because this cannot be proven from other axioms that's put in this way, which sound much simpler. Like, for instance, you have only one line which can contain two given points. I mean, this is something very obvious on the plane. You don't have any kind of negative feelings. Well, why can't we prove it? This is kind of obvious, but this is not, and still needs to be accepted as an axiom. So, the whole theory of parallel lines will actually be based on this particular axiom, and so I will proceed. Now, first of all, terminology. If you have two lines and a transversal, which basically intersects both of them, there's certain terminology about the angles, which are four. There are eight angles, four here and four here. Well, these just among themselves, you know, they're called vertical, and these among themselves are supplemental. We know that. But now, since we have two different lines, we can consider angles like 1 and 2, which are, one of them is called exterior, and another is called interior. So, interior angles are these four angles, which are in between these two lines. And exterior angles are the other four, these and these two. Now, if you are considering this transversal, then the angles on one side are called one-sided, and angles on opposite side, let's say, this one and this one are called alternate angles. So, what we can have here, how can we characterize, for instance, angle one and angle, let's say, three. They are alternate exterior angles. Now, two and four are one sided interior. Two and five are alternate interior. And there is one more, one and two are called corresponding. It's shorter for one-sided, one of them is interior, and another is exterior. They're called, in one word, corresponding angles. Now, what's very important is that parallelism of these two lines is very much related to congruence or supplemental quality of certain angles. Now, if you consider certain angles which are formed by transversal and one of the lines which we consider, and certain angles which are formed by the same transversal and another line, so the corresponding equality or inequality of the angles actually characterizes parallelism or non-parallelism of the lines. Let me just state one very simple theorem as the first, basically, statement in this particular series of statements. Corresponding angles, one and two. So they're one sided, one of them exterior, another interior. So if corresponding angles are congruent to each other, then the lines are parallel. What's important is if the lines are parallel, this is the converse statement, then the corresponding angles are congruent as well, which means that the congruence of the corresponding angles is basically a characteristic property of the parallel lines, which means one defines another. They cannot go, like, there is no such situation when you have two lines which are not parallel to each other, but still congruent corresponding angles. So these are necessary and sufficient conditions for one for another. So corresponding angles are congruent is necessary and sufficient condition for the lines being parallel. Now, not only corresponding angles, not only the congruence of the corresponding angles is this type of characteristic property, also the congruence of alternate exterior two and five is the same type of property. Another example can be that angles two and four, one sided interior, are supplemental to each other. They are together making 180 degrees angle. So there are a few conditions between these angles and each of them can actually serve as a characteristic property of the parallelism of the lines. So let me start from one particular, one and two, two corresponding angles. Now, if they are congruent to each other, then the lines are parallel. Let me prove. Let's consider the lines are not parallel and let's consider they are intersecting somewhere here, if I will extend them sufficiently far enough. So what happens here? Well, let's just consider this point p, point of intersection, and these two points m and n where transversal is intersecting are two given lines. Well, if these lines are intersecting at p, then p and n is a triangle. Now, we all know, and this is one of the prior lectures which I have, that exterior angle of any triangle is greater than any interior not supplemental visit. So if you have a triangle and you have some kind of exterior angle, it's greater than either of interior angles not supplemental visit. And it doesn't depend what kind of triangle it is. This triangle is kind of obvious. Now, if it's a choose, for instance, angle triangle and you consider this exterior angle, it's still greater than any of these two. All right? So this is a property of any triangle. Exterior angle is greater than any interior not supplemental visit. But now let's take a look at the angle one and angle two. And the one is obviously exterior to triangle p, m, n. And angle two is interior angle, which is not supplemental with one. Which means one is supposed to be greater than two. That contradicts our initial premise that the angles corresponding angles are congruent to each other. They cannot be greater or smaller. They're supposed to measure exactly the same. That's the congruence actually is all about. So what we have proven is if these corresponding angles are congruent to each other, then the lines must not intersect on this side. Well, actually, you might say, well, maybe they will intersect on another side of the traversal. Well, this will not be possible, either. So if they are intersecting here somewhere. Now, why is this wrong? Well, for obvious conditions, because let's consider instead of one, let's consider angle one, one, 11. And instead of two, let's consider 22. Same thing. If these two angles are equal to each other, then these angles as supplemental are equal to each other. And now we have exactly the same thing. Now, triangle MPM is a triangle where angle 11 is exterior and angle 22 is interior, not supplemental. So this angle is supposed to be greater than this, which again contradicts their equality, their congruence. So it cannot intersect on this side and cannot intersect on that side of the traversal. So lines do not intersect. We have proven a direct theory. So if corresponding angles are congruent to each other, then lines cannot intersect, which means they are parallel by definition of the parallelism. Now let's prove a converse theorem. What if the lines are parallel? Let's prove that corresponding angles are congruent to each other. Okay. How can we prove it? Well, very easily. Remember the fifth postulate of Euclid. So if some of two angles, one sided interior angles less than a hundred and eighty degree, then the lines will eventually intersect each other. So let's consider 1 and 2. How can we prove that they are equal to each other? They're measured exactly the same way. They're congruent if lines are parallel. Well, let's consider that that's not the case. Let's consider 1 is not congruent to 2. Now what it means that angle 2 and angle 4, now angle 1 and angle 4 are equal to a hundred and eighty degrees, right, since they are supplemented to each other. But 2 is not measured the same as 1, which means 2 plus 4 is not measured as a hundred and eighty degree, right? 1 and 4 are a hundred and eighty. 2 is not equal to 1, right? That's our assumption, which we will basically use to conclude certain things which contradicts our initial premise that the lines are parallel, right? We are trying to prove it from the opposite side. So 2 is not equal to 1. 1 plus 4 is equal to one hundred and eighty degree. That means that 2 plus 4 is not equal to one hundred and eighty degree. Well, okay, fine. That's basically sufficient because it means that 2 plus 4 is either less than a hundred and eighty degree or 2 plus 4 is greater than a hundred and eighty degree. That's what it means that 2 plus 4 is not equal to a hundred and eighty. It's either less or greater. Well, if 2 plus 4 is less than a hundred and eighty degree, then the lines will cross here because of the fifth postulate of Euclid. And that contradicts their parallelism, which we have assumed from the very beginning. That's the condition of our theory. So we came to the conclusion which basically contradicts our initial premise. So our assumption that 1 and 2 are not equal to each other results in some kind of a contradiction. Okay, but this is on this side. What if it's greater than one eighty degree? Well, if it's greater than one eighty degree, that means that if we will consider instead of angle 4, angle 5, supplement to it. So angle 4 is equal to one hundred and eighty minus 5. Now, an angle 22 is equal to one hundred and eighty minus, I'm sorry, angle 2 is equal to one hundred and eighty minus 22, right? So if I will add them together, what will happen? We will have 2 plus 4 here is equal to 360 minus 5 plus 22 in parenthesis, right? Now, this is greater than a hundred and eighty. So this is greater than a hundred and eighty. So what we will do, we will add 5 plus 22 to both sides and we subtract one eighty from both sides. So as we see, 5 plus 22, which means these two on one sided internal angles give a sum which is less than one hundred and eighty degrees, which means it should cross on this side, it should parallel. The lines cannot be parallel basically because they are intersecting. Because of the fifth postulate liquid. So no matter what we do, no matter how we measure, if one is not measuring exactly the same as two, then either the lines will intersect on one side of the transversal or another, which means they cannot be parallel. So we came to contradiction, which means that our original premise that the lines are parallel implies that one and two corresponding angles are congruent each other. So congruence of the corresponding angles is a characteristic property. It's a necessary and sufficient condition using logical language for the lines to be parallel. Now, not only corresponding angles have this type of property, because if you will consider instead of one angle number five, two and five are alternate interior, alternate because on different sides of the transversal and interior because they are in between these two lines. So since these are vertical equal to each other, basically congruent to each other, it means that the theory is exactly the same. If instead of corresponding angles you will consider alternate interior angles. Now, if instead of angle two you consider angle three also vertical and congruent to it, then you can say that one and three, which is alternate exterior angles, have exactly the same property, being the characteristic property of the lines to be parallel. So if exterior or alternate exterior are congruent, then the lines are parallel and if the lines are parallel then all alternate exterior angles are congruent. So there are many different variations as you see of the same theorem applied to different angles. And finally, you can say that some of one-sided interior angles, like four and two in this case, or five and one-two-two, being equal to a hundred and eighty degrees, being supplemental to each other, again that's exactly the same characteristic property of the parallelism, because if these two are equal to a hundred and eighty, since one and four also are equal to a hundred and eighty, then one and two are supposed to be congruent to each other. So they're all very much equivalent to each other and that's why we have how many theorems, like ten corresponding alternate interior, alternate exterior and some of one-sided interior and the same thing, some of one-sided exterior. So yes, it's five different theorems, direct theorems and five different converse theorems, which means if angles are congruent or supplemental, then the lines are parallel and that's five theorems and opposite converse theorems, if lines are parallel, then the angles which we were talking about are either congruent or supplemental to each other. Okay, so this is the base of all the different theorems, future theorems and problems about parallel lines. So parallel lines and transversal angles have the same, have certain property, there are either congruent or supplemental to each other and that's a necessary and sufficient condition for the lines to be parallel. By the way, from which follows a very easy way to solve one of the major construction problems which people might actually face. It's a very elementary problem, but still, I mean any construction problem needs to be somehow explained and basically laid out. Now the problem is if you have a point outside of the line, how to draw a line parallel to this one, which contains this point. Well, there are many ways obviously to do it. Now before, in one of the construction problems which we were solving before, what we could do is we can draw the perpendicular and then another perpendicular here. Two perpendicular will reduce parallel lines. Why? Because these are right angles and two right angles are supplementary to each other and as we know, if some of interior one-sided angles is equal to 100 degrees, then these lines will be parallel with this one as a transversal. But it's a little bit more, I would say, lengthy process to build two perpendiculars. Using the theorems which I have just proven, we can derive with a slightly easier way to do it. Let's just choose any two points here, have a triangle and then using these three sides, have another triangle with this side being common and we will use this radius to draw an arc here, this radius to draw an arc here, so this triangle, this is equal to this, this is equal to this and this is the common side. So these are equal triangles, completely congruent to each other, which means these angles are congruent. But if you consider these angles from the parallel lines standpoint, so these are two lines, this is transversal, so these are alternate interior angles and since they are congruent, the lines are parallel. So this seems to be a slightly easier way to build the line parallel to a given line which contains a given point outside. So you know, you can just use this particular theorem in a real practical problem how to draw a parallel line. Alright, basically that's all I wanted to talk about as far as the parallel lines and transversal are concerned. Obviously the most important is to understand that congruence of certain angles is a necessary and sufficient condition for the lines being parallel. Basically that's it and you can probably use this particular theorem in many other problems and other theorems which you'll find in this particular course. Okay, so that's it for today. Don't forget that the website Unisor.com contains lots of very important educational materials which I would definitely suggest you to use and for parents and supervisors don't forget that you can enroll your students into certain course or set of lectures which have exams and you can examine scores and you can actually decide whether to consider the course completed or basically ask your students to repeat the course to take again the same exam and basically reach the point when all the problems of the exam are solved correctly and that will be the end of the course and then you can enroll to another course etc. Gives you some kind of a very good handle to control the educational process. It's basically kind of homeschooling if you wish and it's a very good aid for homeschooling. Okay, thanks very much and good luck.