 Hi, I'm Zor. Welcome to Unisorification. This lecture would be the last one in series about quadrangles. We will talk about trapezoids. We talked about parallelograms, rhombuses, rectangles, squares. Now it's trapezoid's turn. Well, first of all, where is exactly trapezoid? In the whole scheme of things of all the quadrangles. Now, if you remember, I was talking in previous lecture, I was talking about quadrangle being basically parallelograms which are, in turn, broken into rhombuses and rectangles, and then square inherits both properties of rhombus, which is parallelogram with all equal sides and rectangles, which is a parallelogram with all equal angles. Now, where is in this scheme of things trapezoid? Well, basically trapezoid is here. It's not a parallelogram. In a parallelogram, all four sides are parallel to each other. In trapezoid, two sides are parallel and other two sides are not parallel. Well, actually there is some discrepancy as far as terminology is concerned. Sometimes people might say, okay trapezoid is a quadrangle with two parallel sides, and it doesn't really matter what kind of relationship other two sides have, parallel or non-parallel, which makes parallelogram is kind of a trapezoid with two other sides parallel as well. On the other hand, we can say that trapezoids are actually quadrangles with only two sides, opposite sides parallel to each other, and other two sides are not parallel. It doesn't really matter. I mean, it's just terminology. And speaking about terminology, there is another trick here. You know that Europeans and Americans sometimes are calling the same thing with different names and different things with the same name, like the football, for instance. I mean, in America it's one type of sport and in Europe it's completely different. Or let's say in the decimal number, what's the separator integer part from the fractional part? In America it's dot and in Europe it's comma. So anyway, unfortunately, we have the same kind of a problem with trapezoids. Trapezoid is something which I just explained, a quadrangle with two parallel sides and two other sides are not parallel, something like this. And there are certain quadrangles with all four sides not parallel to each other. Well, that's not the right kind of a picture, but something like this. Okay. All four sides are not parallel to each other. Now, this might be called trapeza, or trapezu, trapezu. Well, unfortunately in Europe, terms are reversed. This is called trapezoid and this is called trapezoid. Well, I'll stick to American side just because I have to stick to something. So let's consider I'm calling the trapezoid this type of figure. These two sides are parallel. Okay. That's some kind of an introduction into trapezoids. And now, as with everything else, we will talk about different theorems, different properties of trapezoids. So I do have my piece of paper where I put all these theorems one after another. Now, what's different, by the way, in explaining the trapezoid, let's say, from something like a rectangle. In case of rectangle, we have inherited certain properties which we have proven before for parallelograms, because rectangle is kind of a parallelogram. Same thing with rhombus. With parallelogram, we didn't have anything before it. So everything for parallelogram, we were just proving one by one. All these theorems and all the properties were unique for parallelogram. We didn't really go any failure prior theorems to refer to. Now, same thing with trapezoids. Since trapezoid doesn't have any predecessors, it goes straight from quadrangle. So all the theorems are basically will be proven in you. And they're really very easy and very short ones. So anyway, number one, basis of trapezoid are not congruent to each other. Well, this is actually following from the definition almost directly. Now, these two sides, which are opposite to each other and parallel by definition of trapezoid, are called bases. This is base and this is base. If you want, this is upper base and this is lower base. And these sides are called legs, if you wish, left and right leg or whatever. Now, why are bases, why they are not congruent to each other? Well, you have to remember the theorem about parallelograms that if two sides, opposite sides of a quadrangle parallel to each other and congruent, then this is a parallelogram, which means all other sides also should be parallel to each other. But we have actually defined our trapezoid as a quadrangle with only two opposite sides parallel to each other and two other not parallel. That's why these bases cannot be the same. One is smaller, one is bigger. And obviously, they can be this way or this way. This is also trapezoid or this way. So, whatever way we can put all these sides as a nonparallel, in a nonparallel position is good enough. So, these are all trapezoids. And that's why the theorem number one was that the bases are not congruent and this is obvious from the properties of parallelogram, as I was just saying. Theorem number two. Okay, it refers to isosceles trapezoids. Now, isosceles trapezoid is the one with legs congruent to each other. So, it might be this position or it might be this position as long as these are congruent to each other. So, either they're both going outside or both goes inside relative to the bottom. That makes top smaller and bottom bigger or top bigger and bottom smaller. I'm talking about bases. Obviously, they cannot go this way because that would make it a parallelogram. So, let's assume we will draw the picture like this. It's just more convenient and more habitual to me. So, isosceles trapezoid, A, B, C, D. And isosceles means B, C is parallel to A, D and A, B is congruent to C, D. That's what isosceles means. So, angles formed by base A, D with two legs are congruent to each other. That's what's necessary to prove. Same thing with these two angles which this base forms with two legs. Okay. How can we prove it? Let's draw two perpendiculars. Now, these two perpendiculars are congruent to each other because we have already proven that if you have two parallel lines, then the distance is basically constant and this distance is measured by the lengths of the perpendicular. Okay. So, these two guys are both perpendicular to the same A, D. So, they are parallel and since it's a distance between two parallel lines, they're congruent as well. So, they are parallel and congruent, which means that, I mean, just incidentally that this is a rectangle. It is important. So, B, C, and M is a rectangle. All right. Now, what do we know about these two triangles? Well, we have two congruent hypotenuses because this is a sosceles trapezoid, right? Now, obviously, these are right triangles because C, M, and B, M are both perpendiculars and also the catatose B, M is congruent to catatose C, M of this triangle. So, these two triangles are congruent to each other, which means these angles also congruent to each other. That's it. Now, how to prove the congruence of these two angles? Well, just looking at the picture, you see that since these angles are right angles plus congruent acute angles of these two triangles, that makes these angles A, B, C, and B, C, D also congruent to each other. So, again, A, B, M is the acute angle which is congruent to M, C, D. And then, to these two congruent acute angles, we have added 90 degrees from this rectangle. So, that makes it also congruent to the sum of these triangles. Okay. So, that's done. Theorem number three in a sosceles triangles, angles are supplementary to each other. Well, we can do it in many different ways. First of all, we know that in this triangle, two acute angles are always sum up to 90 degrees, right? So, if we add another 90, we get a 180, and that makes these two angles supplementary. On another hand, there is, I would say, a little bit more elegant, if you wish, solution, proof actually to this theorem as follows. Now, we know, and again, we have addressed this before, that every convex quadrangle has sum of its angles equal to 360 degrees. Now, since these two angles are congruent, and these two angles are congruent, so the sum of these four angles is 360, but it can be basically divided into two pairs. This plus this, and it's equal to this plus this. Since these are congruent, and these are congruent, then these two sums are supposed to be the same, which means each one of them is equal to 180. Sum of this, same as sum of these two angles. So, that's a little bit more elegant, I would say, but doesn't really matter how you prove it. Anyway, so the angles which lag makes with two bases are supplementary to each other. Okay, number four, medium of a trapezoid is parallel to its basis and equal to half the sum. Okay, I have to draw another one. So, first of all, we are talking about any trapezoid, not necessarily eye substance. Now, medium or mid-segment or midline is a segment which connects two midpoints of two legs. Now, the theory states that this segment mn is parallel to both bases, and its length is equal to half the sum of these two bases. So, if you add this to this and divide by two, you get this one, which looks actually kind of, you know, it looks like it really is half of the sum, because it's in between. This is smaller, this is bigger, and this is in between. So, that's what we want to prove. Now, how to prove it? That's actually easy. All we need is to think about one particular extra line, call it x, and let's consider these two triangles, b, c, n, and m, x, d. Now, these angles are vertical. Now, these angles are alternate interior angles with bc and ad parallel and cd as a transversal, which means they're also equal, congruent to each other. And this segment cn is equal to nd, because n is a midpoint. So, we have angle side angle, angle side angle, which means that this is equal to this, and this equals to this, right? From the congruence of these two triangles. All other elements are congruent as well. Now, what do we have? Let's consider abx triangle. Now, nn is a mid-segment in this triangle as well, because this point is middle of this segment by condition of the theorem, and point n also divides bx into two equal parts. So, n is midpoint of bx. Now, we know about the triangles that the mid-segment is parallel to the base, and is equal in length half of the base. Now, what is the base? But the base is actually sum of two bases of our original trapezoid, because it has ad plus another segment, which is exactly congruent to bc to our upper base. So, the base of this triangle, abx, is actually sum of basis of these, of this trapezoid. So, mn is equal to half of the base of this big triangle of ax, which is sum of the basis. That's why mn is parallel and equal to half of the sum of two bases. And the proof. Okay, what else do we have? Line connecting midpoints of two bases of a socialist trapezoid is its axis of symmetry. So, these two are congruent and these two are congruent. So, x and y are midpoints of ad and bc. Now, how can we prove that points a and d as well as b and c are symmetrical relative to this line? Okay, here's how I propose to do it. Again, whatever, all the problems actually in geometry are going exactly the same way. You have to find some triangles which are equal to each other, congruent to each other. So, I find these two new segments to be convenient in this particular case for the proof. Now, let's consider triangle adx and xcd. Well, obviously, I would like to prove their congruence. How? Well, since this is a socialist trapezoid, then this is equal to this. That's basically the definition of a socialist trapezoid. Now, we have just proven before that two angles at the base of a socialist trapezoid are congruent, each are congruent. It makes these two triangles adx and xcd congruent by side, angle and side. Now, what's the importance of this fact? The importance is that these two segments are congruent to each other as well, which makes xdc an socialist triangle. Now, in the socialist triangle, as we know, median and xy is a median because it divides the bc into paths. It's also an altitude and bisector of the angle and segment bisector, etc. So, it has all these properties. But what we are interested in is perpendicularity. Since it's altitude, it's perpendicular, xy is perpendicular to bc. And that's basically it, because if this is perpendicular and b and c are on the same lengths from the axis lying on the perpendicular to this axis, that actually is a definition of symmetry. Now, same thing here. So, xy, since it's perpendicular to bc, it's also perpendicular to ad as well, because these are parallel. And these two segments are congruent to each other, and that's what makes a and g symmetrical relative to the line xy. Well, that's basically it. Since these points are symmetrical to these, then the whole figure is basically symmetrical. Okay. So, as I was saying, this is the last theoretical lecture about quadrangles. There might be some problems, obviously. And don't forget that every topic actually has an exam associated with it. So, if you go to unisor.com and you sign in as a student, you will be enrolled by supervisor or your parent into some kind of a program where you will have to go through theoretical material like this, end exams, and your score will be visible to your parent or supervisor, and then they can actually decide whether to pass your fail. Well, if you fail, don't despair. Just listen to the lecturers again and try exam again. It's a relatively small number of problems which you have to solve. It's a multiple choice, so just try it and you'll do the best. Okay. Thanks very much for today. That's it. Thank you. Good luck.