 Welcome again to another problem solving session on triangles. Now in this question it's given that O is the midpoint of AB. So this point is the midpoint of AB. So both AB, so this is AB and this is BC, right? Sorry this is CB. So okay so what is the question? Question is we have to prove that triangle AOC is congruent to triangle BOD, AC is equal to BD and third is AC is parallel to BD, okay? So let's try and prove this one. So let's start with writing what is given, okay? So what's given guys? These are the things given. What is given? Simple, O is the midpoint that means AO is equal to OB that's point number one and then secondly CO is equal to OD, right? And why is this? Because O is the midpoint of, O is the midpoint of what AB and CB, okay? This is given. Therefore now let's try to prove. So to prove what do we need to prove? We need to prove that triangle AOC is congruent to triangle BOD, is it it? This is the question, okay? It's the first question, first part. Okay now see how to prove it. So we know side angle, side congruence criteria. So hence we will be using that. So if you analyze this, clearly this angle is equal to this angle. So let us say this is X. So this angle is also X, vertically opposite angle is a bit. So I can say in triangle, in triangle AOC, okay? Let's consider this triangle and triangle BOD. Let's consider the other triangle BOD. So this is BOD and this is AOC, right? So what is already given? We know that KO is equal to OB given and CO is equal to OD. This is also given. Where is it given? See, I had already mentioned this, you know the proofing which has been given. And an angle AOC, AOC is equal to angle BOD. EOD. And why is this? Within brackets you can write vertically opposite angles. Correct? The moment this is done. So what do we conclude? So let me write it now here. So we can conclude that triangle AOC, triangle AOC is congruent to triangle, which one? BOD. Y, S, A, S congruence criteria. So this is proven. No problem in proving the two triangles to be congruent. Now let's go for the second one. AOC is equal to BOD. So AOC will have to be BOD, right? Why? AOC is equal to BD now. Why will that be? Why will that be? So let me write it once again. So AOC is BOD, okay? AOC is equal to BD. So this is AOC here. And this is equal to BD. Why is this congruent part of? Concurrent triangle. So you can write within brackets C, P, C, T congruent part of congruent triangle. So this is point number two. And it is said third one is they are asking to prove AOC to be parallel to BD. And indeed it is. Why? Because if you see this angle is Y, so this angle also is Y, right? This is Y. Why is this? Because of ordered interior. So you can say since angle CAO, CAO is equal to angle which one? CAO is equal to DBO. B, B, O, right? This is true. Why is this true again because of CPCT, right? Therefore, what can we conclude? We can conclude that AOC is parallel to BD. And why is this? Because of alternate interior angle, alternate interior angle, right? Hence all the three parts is congruence, equality and this parallel. All of that is proved, okay? I hope you understood this problem where we applied SAS congruence criterion to prove that two triangles are congruent and their corresponding elements or sub parts of the triangles are equal to CPCT, right?