 Hi, I'm Zor. Welcome to Unizor Education. Today's topic is parallelograms. Well, as usual, we start with definition. So, what is the definition of parallelogram? Well, number one, it's a quadrangle, which means it has four vertices and four sides. Secondly, opposite sides are supposed to be parallel, which means A-B is parallel to C-G, and B-C is parallel to A-G. B-C, A-G. So, that's the definition of a parallelogram. Now, everything else is basically properties. Now, if you wish, theorems about parallelograms. So, I will probably go through certain theorems about parallelogram. These are, I would say, major theorems, and obviously there are many others, but these are something like fundamental properties of parallelograms, and that's why I have included it into this lecture. There might be some exercises or exams, maybe, which I will put into the back page, with different problems, different theorems, etc., but these are fundamental. So, let me just go one by one with these. I have, like, less than 10 of them. Alright. Two angles of parallelogram formed by any one side are supplemental. Let's say this and this. Why are they supplemental? Well, obviously, because since opposite sides are parallel, now we have two parallel and transversal, and we know the properties of the parallel lines and the transversal, that interior one-sided angles are supplemental to each other. Added together, they give 100-80 degrees. That's it. Next, parallel sides are congruent. So, we have this side parallel to this, and this parallel is this. This is given. But I have to prove that these sides also are congruent. Well, that's easy. Let's just draw a diagonal. Now, what can we say about these two triangles? Well, since these are parallel, consider this as a transversal, which means that this angle is congruent to this angle. Sorry, to this angle. I see that something is wrong. This angle. And because they are alternate interior angles, and this one is congruent to this. Now, in these two triangles, this and this, you have one side common, and two adjacent angles are congruent to corresponding angles of another triangle. That's why triangles are congruent, and that's why corresponding sides are congruent as well. Now, but here is an interesting property. So, if it's a parallelogram, then the opposite sides are not only parallel, but also congruent equal size, basically. Now, the converse theorem is also true for, at least for convex quadrangles. So, if we have two opposite sides, this and this, congruent to each other, then this is parallelogram. Now, why? It's actually very simple, because right now, if you consider the same triangles, now triangles are equal, are congruent to each other by three sides. If this is congruent to this, and this is congruent to this. With this common, we have two congruent triangles, which means angles are equal, congruent to each other. But again, we remember that congruence of, let's say, alternate interior angles is necessary and sufficient condition for the lines to be parallel. It's a characteristic property of the parallel lines, that alternate interior angles are congruent to each other. So, that's why, from the congruence of these angles, we derive that the lines are parallel to each other. Same thing with these guys. So, basically, what we're saying is that not only if it's a parallelogram, then opposite sides are congruent. But also, if opposite sides are congruent, then this is parallelogram, at least for a convex case. Opposite angles of parallelogram are congruent. Again, everything follows from the same drawing. Since these two triangles are congruent to each other, that means that this angle is congruent to this one. And since each of these is congruent to the corresponding alternate interior angle, some of these two angles will be congruent to some of these two angles. That's why these angles are also congruent to each other. Well, we go fast. These theorems are really very, very simple. If opposite sides of quadrangle are parallel and congruent to each other, then this is parallelogram. Alright, so, what we have proved before, if opposite sides are congruent to each other, then this is parallelogram. Now, in this theorem, we have only one pair of opposite sides, but condition is they are congruent and parallel to each other. Now, how can we derive from this? And we don't know anything about other pair of opposite sides. So how can we derive that this is parallelogram? Okay, let's just consider again. This is parallel to this and equal in size congruent. Now, this is common. And since these are parallel with this transversal, this angle is congruent to this angle as alternate interior, because the parallelism of these two lines is given. And now, we have congruence of these triangles by side, angle, and side. Side, angle, and side. We use just another property of the triangles, the characteristic property of the triangles, side, angle, side, which is by the way an axiom, it's not a theorem, to conclude the triangles are congruent. And from the congruence of the triangles follows everything else. Now, in particular, this angle would be congruent to this angle, which is sufficient to these two lines to be parallel with this transversal. Okay, now I have so many drawings that I probably have to draw it again. Distance between parallel lines measured along any mutual perpendicular is constant and does not depend on where this mutual perpendicular is drawn. So if you have two parallel lines, well, parallel means they don't intersect. Now, the question is, what if we will measure the distance between these two parallel lines? Well, first of all, let's talk about what the distance is. Well, the distance is between two objects. Let's say between a point and some kind of an object. What's the distance? Distance is the shortest distance among all the different points taken here. Now, if this is also a complex geometrical object, then all these points have to pair with all these points, and the shortest pair is the distance between the objects. Now, let's apply this logic to our two parallel lines. What is the distance between two parallel lines? Well, obviously it's the distance between two points, one of them belongs to one and another belongs to another, and it should be the shortest distance possible. So, let's first pick a point here. Now, we were talking before that perpendicular from a point to a line is the shortest distance. Actually, we did prove this here. Okay, fine. Now, the thing is that what is the shortest distance from this point back to this line? Obviously, it's the same perpendicular because the perpendicular to one parallel line is parallel to another. If this is a right angle, then this is the right angle. Why? Because these are two parallel, this is transversal, and these are two interior one-sided angles which are supposed to sum up to a 180 degree. One is 90, that's why 90 is the second angle. So, the perpendicular is common for both lines. Alright, so basically what we have come up with is that the distance from here to the line is falling to this particular point and from this back to this. So, these two points seem to be on a very, very short distance. You can't really move any one of these points without making the distance longer. So, this seems to be like a distance, at least from this particular point. But what if I will take another point on this line and again measure the distance? Well, obviously we all feel it should be the same. Now, all these words which I was just saying, it's just some kind of an explanation because the proof itself is absolutely trivial. So, if you have two points, two perpendicular lines to this one, then as I was saying, these are usual perpendicular. It's perpendicular to this one, to both lines actually. And why are they equal to each other? Well, because this is a parallelogram, because this is parallel to this and two perpendicular are parallel to each other. We've proved all these points before and in the parallelogram, the opposite sides are congruent. That's why the distance wherever we measure this distance doesn't matter at this point or this point or any other point, the distance will be exactly the same. Too many words for such a simple term. By point of intersections of diagonals of a parallelogram divides each diagonal into two congruent parts. Alright, this is intuitively obvious, but it's interesting to prove. So, if you have two diagonals, then diagonals are divided by this point of intersection into congruent parts. Well, we know what to do basically. We have to include each part into some kind of a triangle and prove the congruence of the triangles. Well, it's kind of obvious. We have B, P, C and A, P, G, these two triangles. They are obviously congruent. Why? Because these two lines are opposite sides of the parallelogram and that's why they are congruent. These two angles, since these are parallel and this is transversal, are alternate interior and these are also alternate interior angles, which are equal considering the lines are parallel. Now, it's this transversal or this transversal. So, the triangles, this one and this one, are congruent by angle-side angle, angle-side angle. From the congruence of these triangles follows the opposite to congruent angles, we have congruent sides. So, this is opposite to this angle and this is opposite to this angle and they are supposed to be congruent to each other. And similarly, this one is congruent to this one. That's exactly what's necessary to prove. Okay, and now, again, an interesting observation. This theorem can be reversed. Conversed theorem is also valid, which means if you have any quadrangle with two intersecting diagonals and the property of this intersection is that it actually divides each diagonal into two congruent parts. So, we have equality of this and this and this. So, this is congruent to this, this congruent to this. Now, the question is, how can we prove that this is parallelogram? Well, easily. Now, this angle is vertical and this is vertical, right? So, now we have side-angle-side, side-angle-side. And that's why the triangles are congruent to each other and that's why we have these two lines, two sides opposite to each other, congruent. Now, very similarly, we can consider this triangle, ABP and this, CGP. And exactly the same thing. This angle is vertical to this one. Sides are congruent and that's why these triangles are also congruent and that's why these opposite sides are equal. And we have already proven that if opposite sides are congruent to each other in a quadrangle, then this is a... Yeah, parallelogram. Okay, and the last but not least, I would say unusual theorem. At least, I don't know, quite frankly, when I first saw it, I was surprised, although it is quite obvious part of the proof is concerned, but let's just consider it this way. You have any quadrangle and you connect midpoints. Guess you will get. You will get parallelogram. Isn't that interesting? I mean, you take any quadrangle, you connect the midpoints and you get the parallelogram inside. Well, why? It's really obvious. The proof is absolutely trivial. Have a diagonal. Now, I hope you remember in one of the prior lectures we were considering a triangle and its mid-segment, which connects two midpoints and we have proven that this mid-segment is parallel to the base of the triangle and it's equal to its half in lengths. Well, here we have exactly the same story. You just use this particular fact, this particular theorem, and what do we see? That this segment is parallel to this diagonal and equal its half. This segment, okay, let me put it this way. So, in the triangle ABC, this segment is parallel to AC and equal its half in lengths. Now, triangle ACG, this is also mid-segment, middle point and middle point, and therefore this is also parallel to the same AC and equal to the same half of AC. So that's why these two segments are parallel both to the same line AC, which means they're parallel between themselves and they're both in lengths equal to half of AC, which means they are congruent. And now we have this theorem which we have proven before about if you have a quadrangle and opposite sides are parallel and congruent, then this is parallelogram. So, again, I would say it's kind of an unusual theorem, although very easy to prove. Well, that's mass. It has certain surprises. At least it was a surprise for me. Okay, that's it for today. Don't forget that Unisor.com contains lots of interesting educational material for self-study as well as for supervised study. And I do encourage parents to really learn how to use Unisor.com site to supervise the self-study of your students, your children, and basically you can enroll them into specific topics or programs. They should take exams and you can actually check the score on these exams and either pass or fail as a student as far as this particular topic is concerned. And it can be repeated as many times as you want until you get the good results. So it's very good for home schooling. Okay, good luck, and that's it for today. Thank you very much.