 Hello and welcome to another session on quadrilaterals. Again we are going to discuss a theorem and then we will prove the theorem as well. So let's first read the theorem. What does the theorem says guys? What does the theorem say? It says in a parallelogram if a diagonal bisects one of the angles, it also bisects the second angle and also the parallelogram is a rhombus. So hence let's first try to construct this parallelogram. So what are they saying? First let's have a parallelogram. So what is the parallelogram guys? So parallelograms we know the opposite sides are equal and parallel. So let me draw this. So this is the desired parallelogram let's say a, b, c, d. So what is b, b will be here, c point comes here. So we have to first join a diagonal. So let's create a diagonal. Let's say b and d I join. So b, d is the diagonal. Now it also bisects. So it says if a diagonal bisects one of the angle that means if this angle is equal to this angle that is angle a, b, d is equal to angle d, b, c then the other two angles are also bisected. Other angle that is angle d is also bisected. Okay for that we have to first verify it. So let's do one thing. First let's create constructed. So this is a rough you know diagram in the sense here if you measure the angles will not be equal. So let us say a, b, d or rather d, d, a or this is c, b, d is equal to 30.96 and d, b, a is equal to 40.6. So if you can see these two angles are not equal here. Not equal. Why? Because the construction is such not every parallelogram will have a diagonal bisecting the angle. So let's first construct that. How to construct that is also interesting. Okay, so let me delete this part. So what I'm going to do is I am going to delete everything here. So let me just delete this. So deleted, deleted, deleted. So I'm just taking away all that one. Yeah. So what we'll do is we'll construct such a parallelogram where the diagonal is bisecting one of the angles and then we will try to verify first using this tool jujibra that whether the other angle is also bisected. And if it is true, then let's try to prove it. So let's try to first draw. So what I'm going to do is I'm going to draw a line segment first. So let's say a, b. Okay. And let us say I draw another line a and here let's say or rather segment I will drop. So what I'll do is I will try to draw a segment. So let us say a to this part c. So let me rename it to d because I need this as point d. Okay, so I have renamed it. So this is a d and b. So I'm going to bisect this triangle, sorry, angle d, a, b so that I get the first corner of the parallelogram if you can see that. Okay. So let me bisect it. So how do I bisect? There's a tool available angle bisector. So here is angle bisector. So I bisect that. Okay, so I'm not using the regular construction mechanism where you use the compass and draw an arc and all that. So clearly this is the bisector. Okay. But if you see now, let us try to find the opposite vertex to a that is I'm trying to locate point C. And you will see why the second line in this question is given. So if I try to locate, let's say a point from a line. So I've drawn a line parallel to AB passing through D. And let us say if I consider this point to be the point of intersection C, so BC, let me join this line BC. Okay, so BC. Now you can clearly see that ABCD is not parallelogram. Is it? So that's where the catch is you can see the second line. But what does it say? It says also the parallelogram is a rhombus. That is a hint that this parallelogram where the diagonal is bisecting one of the angle and the other angle as well. So both the opposite angles are being bisected by a diagonal. That's only possible in the case of rhombus. And that's actually true. So first, let's draw a rhombus and then let's verify that. Okay. So how do we create a rhombus? So let me first delete everything. So again, what I'm going to do is I'm going to delete all. Okay, so we are going to create a rhombus. And then, okay, so how do I create a rhombus? So rhombus, what is a rhombus? Rhombus has all sides equal. Is it? And it's a parallelogram as well. So what I'm going to do is I'm going to draw a circle passing through this point AB. Now, why did I do this? Because now the second point D which I'm going to locate must be on this circle. Why? Because whatever point I take, let's say I take this point C, but I have to rename it D because the reason was I have to name it in that order. So let's say D is point on the circle. Now, clearly AD, AD will be equal to AB or not. Why are they equal? Because they are the radii of the same circle. And I needed two adjacent equal sides for a parallelogram, right? Then only I can say that it's a rhombus. So now I have got one corner of the rhombus with equal sides, one angle with two equal arms, right? Now, if I draw parallel lines from both D and B, then I'll get a rhombus. So it's very easy. Let's draw a parallel line from D and this line is parallel to AB, let's say. So this is, you know, this is going to give me the point C. Now, from B also I'm drawing a parallel line to AD and this is point C. So very clear. I think this is clear construction of rhombus. Okay. So I have located on point C wrongly. So this is point C. Perfect. So ABCD is the rhombus. Now I don't require, let's say these circles. So let me take it away. I also don't require these lines. I just needed the point C. So let me take this away as well. So gone and I'm simply going to join this DC like that and CV BC. So ABCD is my required rhombus. Now, what I'm going to do is I'm going to join the diagonal AC joint. Now let's measure these two angles, which are BAC. Okay. So this is BAC 33.13 and this side and this side 33.13. Perfect. So you can see this is bisecting, right? So the diagonal AB is bisecting. Sorry, diagonal AC is bisecting the angle A. Let's measure, try to measure this one or the other one. And can I? Yes, I can. And I can do from here. So let's say this one and this one is 33.13. Perfect. This one and this one. I'm sorry, I chose the wrong point. So this one and this one 33.13. So you can see all the angles being bisected are all equal, right? So alpha, beta, gamma, delta, all are 33.13. So it's very clear that in a rhombus and angle bisects both the angles, both the opposite angles. Okay. Now let us try to verify by changing the location of B. So C, I'm changing D without changing this length. So if you see the point D is moving on that circle. Let me show that to you. So let's say I am bringing the circle back. So I'm changing point D's location such that AD is always equal to AB being the area of the same circle. Okay. So the rhombus is formed at different, different locations. You can see that. And as the rhombus are being formed, you can see the angles are always equal. That means the diagonal AC is bisecting angle A and angle C. Now the question is that proves that this is true for all such parallelograms. Or rather you have to prove that such thing exists only in rhombus. So let's try to do that. First, let us take away those things which we do not require. So these values I don't need. So let me take this, take them away. Okay. In this, I don't need any value. So let me take that up. Okay. Now let me also take this circle away. I don't need this circle. Okay. Now. So what are we going to do here? So we are always going to use the help of, we are going to use or take the help of congruences. So let's start with this. So given, what is given guys? So we have to write that ABCD is a parallelogram. ABCD is a parallelogram. Okay. Now if ABCD is a parallelogram and angle DAC is equal to angle BAC. Let's say this is given and let us assume that is this is X. So let us say this is X and this is X and clearly then AC is bisecting angle A to prove what do you need to prove to prove part A. There are two parts to the question. One is that angle DCA is equal to angle BCA. So that means if this is angle Y and this angle is let's say Z then you have to prove that Y is equal to Z. That is Y is equal to Z. And secondly, second part is ABCD is a rhombus. Is a rhombus. Okay. So I hope you got the question clearly. Now let's try to prove proving is easy. I told you we have to go for congruences. So let's take the proof. So how to go about it? Clearly, if you see how do I prove that? Okay. Now if you look at this particular thing, this is a parallelogram. So don't you think that since, since AD is parallel to BC, correct? AD is parallel to BC because it's a parallelogram. So what will happen? Angle X will be equal to angle Y. Is it Y? And you can write the reasoning alternate interior angle. Yes or no guys. Similarly, BC is parallel to AB. And therefore, therefore what? Again, so this to this angle, let me just change or take another color. So this is going to be equal to this and this is going to be equal to this. Clearly, why is that? Because interior angle, right? So angle X is equal to angle Z. Alternate interior angle. Fair enough. I think there is no issues in this, no problem. So hence, we can say angle Y is equal to angle Z from these two above statements, right? And the moment angle Y is equal to angle Z, we can say that AC bisects, bisects. In fact, this B should be the small letter. Let's also maintain the rules of English. So AB bisects, AB bisects, angle C as well. So hence, if AC bisects angle A, it does bisect angle C as well. Also, now if you see, in triangle, let us say in triangle ABC, let's take this triangle. Now we are going for part B. So this was for part A. So we are going for part B now. So in triangle ABC, clearly, angle BAC is equal to angle BCA, right? We just proved above. Yep, X is equal to Y, X is equal to Y. Is it C? X is equal to Y. Since you can say X is equal to Y. Therefore, the triangle is as, I saw so this triangle opposite angles are equal. Then opposite sides are also equal. They are trying to, therefore, AB is equal to BC, right? AB is equal to BC. Now if AB is equal to BC, now what is AB equal to, if you see, since this is a parallelogram, since ABCD is a parallelogram, ABCD is a parallelogram, then AB is equal to, what is AB equal to? AB equals to CD? Yes. Now we just proved that AB is equal to CD, a BC also. So that means AB is equal to CD is equal to BC. The three sides are equal and we also know that BC is equal to AD. So hence, we can just simply add BC is equal to AD. So that means a parallelogram with equal sides, right? Parallelogram with equal sides, with equal sides, sides is a rhombus. Is that guys? So we did prove that this particular parallelogram, parallelogram is a rhombus, okay? So I hope you understood the proof. We will take up another theorem in the next session. Thank you.