 Hi, I'm Zor. Welcome to Unizor Education. I will continue talking about quadrangles, and today's topic will be rhombus. Rhombus is basically a kind of a parallelogram, parallelogram, which I have actually talked about in the previous lecture. Now, what kind of parallelogram is it? Well, the one which has all sides congruent to each other. So basically, it's like basically taking a parallelogram, but not just any parallelogram. You know that generally speaking, the opposite sides are congruent to each other. That's the general parallelogram. But the one which we are talking about is the one which has all sides equal. So these sides, all four sides are equal to each other, congruent to each other, equal in size basically. Okay, so that's the definition of the rhombus. Now, since the rhombus is parallelogram, and it also has some specific quality that all sides are congruent to each other, all the properties of the parallelogram are supposed to be true for the rhombus as well. Now, I will just mention all these properties because they were discussed and proven in the previous lecture. So I'll just mention them here. That's why I have this piece of paper. Okay, two angles of parallelogram formed by any one side with both neighboring sides are supplemental. So these two angles are supplemental to each other. Now, again, this is a property of the parallelogram in general, which means any rhombus also has this property. And I'm not going to prove it because it's proven for parallelograms. Two angles form parallel sides are congruent. Okay, now from the definition of the parallelogram, and the definition was that opposite sides are parallel to each other, we have proven that opposite sides are also congruent to each other. Well, this is a property of the parallelogram. Now, in rhombus, we bring it further by actually defining a rhombus as not only these four but these two as well, and all four of them are exactly the same lengths. That's the definition of the rhombus. But still, the property of the parallelogram is that the opposite sides are congruent to each other, this to this and this to this. Next, if two pairs of opposite sides are congruent, then this is parallelogram. So, we have proven that if this side is congruent to this and separately this to this, then it's parallelogram. Well, in rhombus, we have actually defined the rhombus as four sides being congruent. So, that's why rhombus is obviously a parallelogram. Opposite angles are congruent. So, this angle is congruent to this. It's true for a rhombus as well, because rhombus is a parallelogram. If two opposite sides are parallel and congruent to each other, then we don't really care what kind of relationship between other sides, it's already parallelogram. Okay, in rhombus, we have a different quality, but nevertheless, whatever we have stated for parallelogram is definitely true for rhombus. Distance between parallel lines measured along any mutual perpendicular. Okay, this is a property which was stating that if you have two parallel lines, then no matter where you measure the distance, it will be the same. I proved this for parallelograms, actually, because I was using parallelograms, but nevertheless, I just mentioned it here because it was in the previous lecture. A point of intersection of diagonals in a parallelogram divides each diagonal into two congruent parts. Okay, so if you have two diagonals, one and two, this point divides each diagonal into two parts congruent among themselves. And this is true, obviously, for a rhombus as well, because rhombus is a parallelogram. If a point of intersection of two diagonals in a quadrangle divides each diagonal into two congruent parts, then it's a parallelogram. Well, it's actually a converse theorem which states that if you have two diagonals divided by intersection point in each diagonal divided into two congruent parts, then it will be parallelogram. Again, it's a property of the parallelogram. It's not really much related to the rhombus because we will actually prove something else, which is much more interesting about perpendicularity of these. Okay, and the last but not least, sequentially connected midpoints of any quadrangle, sequentially connected midpoints, form a parallelogram. All right, this is just a property of parallelograms and doesn't really matter for rhombus right now. I just mentioned it as a repetition because rhombus is a kind of parallelogram. But now, since we have some additional property with rhombus, which is different from general parallelograms, namely, all four sides are congruent to each other. We have certain additional properties of the rhombus which are not applicable to general parallelograms. And here they are. I have a couple of theorems here. First of all, diagonals of a rhombus are perpendicular to each other. Okay, this is very important and this is a particular property of the rhombus, that these diagonals are perpendicular to each other. Now, how can we prove it? Well, actually, it's quite obvious. Since it's rhombus, let's put some letters around it. Since it's rhombus, sides A, B, and B, C are equal to each other, which means A, B, C is equilateral triangle. Now, from the properties of the parallelogram, which I was just repeating, we know that this is midpoint of diagonal AC, which means, let's put this one to M, which means B, M is a median of A, B, C triangle. Now, the median in an isosceles triangle, do I state it equilateral? No, it's isosceles triangle. I'm not sure. Anyway, in the isosceles triangle, median B, M is also an altitude and angle bisector. Remember that property of the isosceles triangles. Now, in this case, we are interested in the fact that this is altitude, so this is right angle. And that's exactly what's necessary to prove, that diagonals are perpendicular to each other. Okay, that's the first theorem. Second is that diagonals are angle bisectors. These angles are equal to each other. And these angles are equal to each other. And these and these. Again, it's very easy to derive it from the fact that A, B, C is isosceles triangle. Because again, median is angle bisector as well. So basically that's the proof. It's exactly the same as the perpendicularity of the diagonals. We just use a different property of the median in the isosceles triangle. One property was it's an altitude. Another property is it's an angle bisector. So both are needed. So diagonals are perpendicular to each other and are angle bisectors. Okay, next each diagonal of a rhombus is an axis of its symmetry. Now, do you remember what axis of symmetry actually is? So if you have a line and you have two points on opposite sides in such a way that this is a perpendicular and these two segments are congruent to each other, then these points are called symmetrical relative to this axis. Or reflection one of another. It's like a mirror reflection basically. Now, what this theorem actually states, that any diagonal is such an axis of symmetry. So if you take this piece on the right of the BD and turn it over, it will coincide with this piece of a rhombus. Now, how can we prove it? Well, basically we have already proven it because we have proven that diagonals are perpendicular to each other. Which means since this is right angle and also this piece of diagonal is congruent to this piece because as we know from the parallelogram property that diagonals are intersecting in the midpoint. So the point C is symmetrical to point A because they are on the perpendicular line to the axis BG and on equal distances. Now, if A and C are symmetrical, now B and G points, they are lying on the axis itself which means transformation of reflection actually transfers them into themselves. They are symmetrical to themselves. So if you have an axis of symmetry, this point is symmetrical to this but this point is symmetrical to itself. Which means that the whole rhombus, since A and C are symmetrical, B and D are on the axis of symmetry. That means that the whole segment AB would be symmetrical to BC and segment AD would be symmetrical to CD which means that the rhombus is symmetrical relative to this diagonal. And obviously this diagonal has exactly the same properties. There is no difference. Okay, next theorem. Point of intersection of rhombus diagonals is equidistant from all its four sides. Alright, now what does it mean? It means that the distance from here to this side and to this side and to this side and to this side is exactly the same. Now, how can we prove it? Well, if you remember, any diagonal of a rhombus is also an angle bisector. Now, from the properties of angle bisector, which we did discuss before, any point on the bisector is equidistant from both sides. Which I can prove actually very easily because these two angles are congruent and this is a common phytonus. Right triangles are congruent by phytonus and the cute angle and that's why these legs are congruent as well. So, that was done before. I mean, the proof of this was done before, so I can just use it. Again, this diagonal has a property that every point on it is equidistant from these two sides of an angle ABC. It's bisecting. Now, but it's also bisecting the angle ABC which means, again, every point on this diagonal is equidistant from these two sides and considering that the same, that this diagonal is equidistant from these two sides and this piece is equidistant from this. So, the point which is in the intersection has all the properties of both diagonals which means from this, the distance is the same from here to here because it lies on this diagonal but it also from here to here because it lies on this diagonal and from here to here because it lies on this and from here to here because it lies on this. Since this is an intersection point, it's equidistant from all four sides of the rhombus. Because, again, if the point belongs to one set of points which has certain properties, and at the same time it belongs to another set of points which has certain properties, then if it belongs to an intersection of these two sets, it has both properties. OK, next. Sequentially connected midpoints of rhombus form a rectangle. OK, so if you have midpoints here, here, here, and here, and you connect them, you will get rectangle. Now, rectangle is actually subject to the next lecture. By definition, rectangle is a parallelogram, also parallelogram, which has additional property that all its angles are congruent to each other. And, obviously, it can be very easily proven that they're supposed to be 90 degrees, right angles. So in any case, how can we prove that? Let me think. Oh, it's very easy. You see, this line, let's call it M, N, P, and Q. So the segment NP in the triangle BCG is a mid-segment, the one which connects to midpoints of two sides. And it's parallel to the base, BG, in this case. And, by the way, it's equal to half of the base, if you remember that theorem, which we did actually go through before. So NP is parallel to BG, and in length, it's equal to half. But same thing, if you consider triangle ABG, M, Q will be also in its segment, which means it's parallel to the same BG and equal in length to the same half of BG, which means these two are parallel to each other, since they are parallel to the same BG. And they are congruent because their length is equal to half of this on both sides, which means M and P, Q is parallelogram. OK, that's easy. Also, since it's parallelogram, and we know that this is parallel to this, and this also is parallel to this, but we can do exactly the same logic and derive that Mn is parallel to diagonal AC, and PQ is also parallel to AC. So Mn and PQ are also parallel to themselves. But now, let's think about what's the angle between Mn and Mp? Now, you remember it again, you remember in one of the prior lectures that if you have two angles with mutually perpendicular sides, then these angles are supposed to be either supplemental or congruent to each other. So if you have one angle and then another angle, so this is right angle, and this is right angle, then this is equal to this. So here we have exactly the same situation. Mn is parallel to AC, and P is parallel to Bd. So we have two angles, M and P and B and C, with mutually parallel sides. So actually, it's not perpendicular. I have another theory here. If two angles have parallel sides, then they are also equal or supplemental to each other. OK, so it's not the perpendicularity of the sides. It's parallelism between the sides. Since Mn is parallel to AC and P is parallel to Bp, then this angle between them is equal or supplemental to this. But we have already proven that the angle between the angles of the rhombus is the right angle, 90 degrees, which means this is also 90 degrees. And since it's a parallelogram, then the second, the next angle, which is supposed to be supplemental, is also 90 degrees, and this is 90 degrees, and this is 90 degrees, so all of them are 90 degrees. All of them are right angles. And that's why this is a right angle. And that actually includes all the specific properties of the rhombus, which I wanted to go through. Don't forget that unison.com website contains lots of different educational materials. And it's very useful for parents who want to supervise the educational process of their children by enrolling them into this or that specific topic in the course and checking the exams, checking the scores, and making a decision about whether to pass or to fail the student for this particular topic. And if he or she fails, all it takes just to go through it again and again and repeat the exam until you will be perfect. And parents and supervisors will be satisfied with your score. OK, good luck, and thanks very much. Next lecture, I will talk about rectangles and more details. Thank you.