 Hello friends, welcome again to another session on triangles. As we discussed in the last session, we are now going to, you know, solve more problems on the learned concepts. So, in this question, it's given that angle BCD is equal to angle ADC. So, let me start. I'm saying BCD, BCD, angle 1, which is marked as angle 1 here, is equal to angle ADC, ADC, which is marked as angle 2. And ACB, where is ACB? ACB, you can see ACB, angle 3 is marked like angle 3, is equal to BDA. So, BDA, clear? It's given. Prove that AD is equal to BC. AD is equal to BC. Now, this appears to be fairly simple. And angle A is equal to angle B, we have to prove it. And I will add something from my side. So, usually, this is the practice which you should follow, that let's say there is a given problem. Try to build on that. Can we make it a little complicated? Now, you'll ask why unnecessarily complicate the matters. But that's fun. That's what mathematics is all about. So, let's see if we can complicate it by doing what I can, I am thinking of this that, and in fact, this is a property later on you will study that if I join AB, in this case, if at all we prove whatever is asked for and then I join AB, then you will be astonished to know that CD, CD is parallel to AB. Will that be right? So, that's what I am exploring over and above what's given in the question. I will explore this. So, before that, let's first establish what's given and then we can explore further. So, what was asked for? We have been asked to prove that AD is equal to BC. And now that you are equipped with all the congruence criteria, you can do it very easily. So, as we have been doing, let's write first the customary steps. So, given what is given folks given, I know most of you hate this part, but actually gives a lot of clarity to the mind also, because the moment you write all of this, it becomes easier to manage the data and hence navigate to the solution. So, what's given? So, it's I personally like writing all these steps because it gives a lot of clarity to my mind. So, what's given? What's given? In the given figure BCD. So, let's write that angle BCD. And in that process, I also identify those elements. So, BCD, BCD is equal to ADC. So, angle ADC. That's very, very good. Now, another one is angle ACB. ACB is equal to angle BDA. So, don't just copy also, you know, get that clarity from, you know, with one figure. So, trace it out. So, the moment I say ACB is equal to BDA, trace it out. ACB and BDA, yeah, which is also highlighted as three and four. Fair enough. And we have to now prove that only these two things are given by the way. Okay. Now, we have to prove that AD is equal to BC. Now, whenever we have to prove two geometry elements to be equal, what do we need to? If somehow we prove that they are congruent part or corresponding parts of congruent triangles, our job is done. That's what I will look for. So, where is AD? Sorry, AD and BC. Which two triangles are the triangles where AD and BC are found out? So, AD is a part of ADC, no doubt about it. And BC is a part of BCD. So, hence, if we somehow prove that these two triangles are congruent, then by CPCT, we can always say that AD is equal to BC. Let's do that. So, let's take triangle ABABC, ADC first and triangle A, which one? Not A. So, this is what I was talking about. It gives a lot of clarity if you have written down these steps of the given fact. So, ADC will correspond to BCD. So, let me write BCD, match it. A has to be equal to BS, D has to be equal to CS and C in other sense, other angle C has to be equal to angle D. So, this is what we are going to achieve. So, in triangle ADC, what do we see guys? We have these two angles anyways given. So, what is angle 2? So, for ADC, 2 is important. So, I am writing angle ADC anyways. So, in triangle ADC, angle ADC is equal to BCD. It's given. I don't need to do much here, BC, BCD, ADC. They are the same. Correct. Now, one angle I have achieved which is equal and you can also see the side CD or in this case, CD is equal to DC, right? In these two triangles, CD is equal to BC, right? CD was part of ADC and both are equal anyway. So, this is common side. So, no problems here. Common, right? What else? What else can I say? Now, can I say that angle ACD is equal to angle B, D, C, S, Y because if you see 3 plus 1 and 2 plus 4. So, let's write, right? So, you can see angle 1 is equal to angle 2 given, right? And angle 3 is equal to angle 4. This is also given. So, if I add both of them one by one. So, angle 1 plus angle 3 should be equal to angle 2 plus angle 4, isn't it? Now, 1 plus 3, what is it? Angle ACD. And the second angle is BDC, B, D. See, just by observation, you can figure this out. Now, so hence, third here, I can always write angle, angle ACB is equal to angle B, B, C, correct? And hence, what do we learn? We know now by, by what? Which criteria guys? A S, A congruence criteria, you can say, you can say that what angle or triangle A, A what ADC, ADC is congruent to triangle B, ADC, so BCD, right? Now, the fallout will be that you can write the repercussions. repercussions are what, what was to be found out? So, AD, so AD clearly, so AD will be equal to, AD will be equal to BC, correct? And why is that? This is called CPCT, right? And also angle A, angle A, which angle A? So, CAD, so let me write fully. Let me write CAB, CAD is equal to B, BC, angle B, BC, right? This is what we got. Both of, both of these results we obtained. Now, as I said, we will just try to complicate this a bit. What is that complication? So, I have drawn AB, joined AB, okay? So, the moment I joined AB, can we prove that, can, can do you think that AB is parallel to, so this is an extra exercise, AB is parallel to CD possible, possible? This is a question. So, for that, what do we need to do? Let's see. So, AB, being parallel to CD will give me what, this should be angle one, and this should be angle two, right? Is that so? So, that's what we are trying to achieve. Tell me, is that so? Is that so? Before that, a few things. What all? Angle one is, okay, do we know that angle one is equal to angle two? Yes, we know that. Is it angle one is equal to angle two? Why? Because of this, this statement, okay? This statement says that angle one is equal to angle two, because angle one was BCD and angle two was ADC, right? See, this one. So, hence, I know that angle one is equal to angle two, okay? So, guys, if angle one is equal to angle two, then, right now, we don't know whether these are one and two. So, let's do one thing. I am writing this as five. So, let me write this, because we don't know, we have to prove that these are one and two. So, hence, let's call it X and let's call it Y. Okay. Now, and let me call this point O. O, right? So, tell me, will do you think AO is equal to OB? Will AOB equal to OB? Yes, it will be Y. See, we already know that AD is equal to BC. How? We just proved it here. Here, see, we proved it. So, AD is equal to BC. Also AD, right? So, you can say OD is equal to OC. Why? Because angle one is equal to angle two. So, hence, OCD becomes an isosceles triangle. Interesting. Now, if you subtract these two, AD minus OD. So, AD minus OD is equal to BC minus OC. What will this yield? This will give you OA is equal to OB. The moment you say that OA is equal to OB, what do you conclude? You conclude X is equal to Y. So, this will imply X is equal to Y. Isn't it? Now, if you consider this angle to be Z. So, clearly, this also is Z. So, let me use this color here so that I can differentiate. So, this angle is Z. Right, guys? So, hence, in the two triangles, X plus Y plus Z is 180 degree. Can I write that? And so, hence, 180 and 1 plus 2, angle one plus angle two plus Z is also 180. In the second triangle, OCD, OCD, check. In the second triangle, OCD, 1 plus 2 plus Z is 180 and OAB. X plus Y plus Z is also 180. So, now, this Z and this Z will go. Angle one was equal to angle two and X is equal to Y. So, can I not write 2X is equal to 2 angle one. Right? That means X is equal to angle one and done. So, X is equal to one. The moment X is equal to one and you can now conclude here. So, let me draw some boundaries. Yes, I get some space. So, what do we conclude? If X is equal to angle one, then clearly AB is parallel to CD. This is interesting. That means A, B, B, C is a trapezium, folks. Trapezium. Right? That's interesting. So, we started from somewhere, we ended up somewhere else. So, that's how so many insights can be drawn from one given problem. So, that's how you should be solving more problems. Whatever is given, that's fine. You arrive at it, arrive at the solution and the result. But also try to see whether we can really do something more about it. What else? There are many other things which you can prove from here. For example, this is also an isosceles trapezium. Isosceles trapezium means the nonparallel sides are equal. Also, the base angles are equal. So, if you see, let me take this color and yes, so this angle is equal to this angle. This is equal for isosceles trapezium. Also, this angle is equal to that angle. So, there are so many things which you can derive from one simple given question. That's how you should be solving problems, my dear friends. So, let's take up another problem in the next session and let's try to make it more interesting. I hope you learned something from this. Let's meet again. In the next session, bye-bye. Take care.