 Hi, I'm Zor. Welcome to Unizor Education. I continue my course of lectures about different quadrangles, and today's topic is rectangle. Now, before, I talked about parallelograms and rhombus, which is a specific kind of parallelogram with all equal sides. Now, rectangle is defined as also a specific type of parallelogram, which has additional property, namely the angles of this parallelogram are all supposed to be congruent to each other. Now, in the general parallelogram, we have only opposite angles congruent to each other. These are this to this and this to this, and they are different between themselves. They are actually supplemental. Now, if we are talking about rectangle, then all four angles are supposed to be congruent to each other, and since these two angles in a parallelogram are supposed to be supplemental and equal to each other, as I said, they are supposed to be equal to 90 degrees, which means they are all right angles. All four of them are right angles. Okay, so that's basically the immediate property which we can derive from the definition of the rectangle. So, the definition has the only one particular requirement. All angles are supposed to be congruent to each other. But considering we know the properties of the parallelogram, we have basically proven a theorem that all angles of a rectangle are right angles. And how did we prove it? Well, based on the fact that these two angles in a parallelogram are supplemental. Since they are equal, 180 degrees divided by 2, that would be 90. Okay, so at the same time, I have to mention that rectangle is parallelogram. It's a specific type of parallelogram. Now, that means that all the properties of the parallelogram are true for the rectangle. And as I will be trying to prove certain new properties of the rectangle, I will definitely relate to already known properties of the rectangle, which basically follow from the fact that this is parallelogram. Like for instance, I just used the property of the parallelogram that these two angles are supplemental. And that's how I have proven that all four angles are right angles. So, let me go through other theorems. And again, I will always refer to certain known properties of parallelogram. One is, so to speak, converts. If you know that this is a parallelogram and only one angle is equal to 90 degree, then all other angles will also be equal to 90 degree, by the way. But what's important is that this is rectangle, which means all angles will be equal to each other. Now, how can I prove it? Again, I know that this is parallelogram. So, if I have proven that some geometrical figure is parallelogram, and I have noticed that one particular angle, let's say this one, is at the right angle, then from the properties of the parallelogram, I know that another angle is also 90 degree, and then another, and then another. And that's why all of them are 90 degree. And that's why it's not just any parallelogram. It's the parallelogram with all angles equal. And that means it's a rectangle. Next. Diagonals are congruent to each other. Okay, how can you prove that these diagonals are congruent to each other? Well, actually, it's quite simple. These triangles, A, B, C, D. Let's say A, B, D, and A, C, D. They are congruent. Why? Well, they have congruent legs, and another leg is shared by both of them. And since these two triangles are right triangles, this angle is 90 degrees and this is 90 degrees. So we have congruence of two triangles, A, B, D, and A, C, D, by two legs, two cassettes. Since they are congruent, the hypotenuses are congruent as well. So in the rectangle, both diagonals are congruent. Next. Sequentially connected midpoints of all sides form a rhombus. Now the previous lecture was about rhombus, and the rhombus is defined as a parallelogram with all sides congruent to each other. Well, let's see what will be if we will connect midpoints here, here, here, and here. Now, why is it rhombus? Well, think about it this way. Consider triangle A, B, C. A, C is the base, and let's say M, N, and M, N is a mid-segment because it connects two midpoints of two sides, A, B, and B, C. And we know from the theory of triangles and parallel lines, actually, that segment M, N is parallel to A, C, and P is mid-segment equal in size to half of A, C. But now consider A, C, D triangle. It also has the mid-segment P, Q, which is also parallel to the same A, C, and also equals to its half. Now, triangle B, C, D, and P is mid-segment, which is parallel to B, D, and equal half of its size, same as M, Q. But now, considering B, D, and A, C are congruent to each other, which I have just proven five seconds ago, it looks like M, N, N, P, N, P, Q, and M, Q all equal to half of either this or that diagonal, and diagonals have the same size, so these sizes are all equal to each other and equal to half of diagonal. So M, N, P, P, Q, and M, Q all are congruent segments, which makes M, N, P, Q a rhombus. All right. Now, here is an interesting kind of auxiliary theory, if you wish. If you have a line and two points on it and you draw perpendicular in both sides, and you cut exactly the same height, then this is rectangle. Now, why is this rectangle? Well, two perpendiculars are to the same line and are parallel to each other. We have learned that from the previous lectures. And they're also equal in lengths, because that's the way how I constructed these two points. I have exactly the same lengths. So these two sides of this rectangle are parallel and congruent to each other, which makes it parallelogram. Now, why is this parallelogram a rectangle? Well, obviously because this was perpendicular by construction, right? So that's why this is rectangle. All these angles are 90 degrees. Okay, so this is proven. Just keep it in mind, and we will use it in the next problem. So again, two perpendiculars of the same size to the same line make a rectangle. So what is the next theorem, which is I'm going to prove? The line to which we draw through midpoints, opposite midpoints of a rectangle, is its axis of symmetry. So this side is symmetrical to this. Well, obviously the line through these midpoints will also be the axis of symmetry. Now, how can we prove it? Well, think about this way. A, D segment, A, M, and P, D are two perpendiculars of the same size. Now, why are these the same size? Because these are midpoints, so the height is exactly half of the size of the rectangle, and these sides are congruent to each other. So A, M, P, D is also a rectangle, which means that this is the right angle. So is this on this side. So let's consider points A and B. They are on the perpendicular, since these two angles are 90 degrees, so A, B is perpendicular to M, P. And also the lengths of these two segments, A, M, and M, B, is the same because M is a midpoint of A, B. So A and B are symmetrical by definition of the symmetry relative to the axis M, P. They are on the perpendicular to the axis and on the same distance from it. Same thing, C and D. And that's why the whole rectangle, if you will reflect it relative to M, P, B will be in A and C will be with D. So the whole rectangle is symmetrical relative to this reflection. And obviously, relative to this reflection, this line as well. Okay, that basically concludes my specific properties that I wanted to talk about, about rectangles. Not many of them, because most of the properties rectangle inherits from the parallelogram and it adds something new primarily related to the quality of the diagonals and symmetrical points, etc. And the fact that these are all right angles at the vertices of this rectangle. So the only thing which is left actually is, for the next lecture, is squares in the area of quadrangle. So have a couple of words about squares. Anyway, that's it for today. Don't forget that unizord.com website contains this and many other lectures. And I do encourage parents and supervisors to use this particular site to control the educational process of their students. I mean, obviously, students can go and just, you know, listen to any lecture and do whatever they want with the website. But for the parents and supervisors, there is a way to really control the educational process. You can enroll your student into a particular program. You can check the exams, the scores, and you can pass or fail the student based on the score. And failure will, you know, it will cause probably the repetition of the lectures and the problems, etc. Going through exams again until everything is perfect. Okay, so that's it. Thanks for today. Thank you very much. Bye-bye.