 Hello and welcome to another problem-solving session on congruent triangles. In this question It's given that AC this is AC AC is equal to AE AB is equal to AD. AB is equal to AD. Okay, looks like this is a nice ausilist triangle I don't know whether that is going to be useful. Let's check BAD is equal to EAC angles BAD so it's shown here BAD is equal to angle EAC We have to prove that BC is equal to DE So first find out BC. Where is BC? This is BC. This is BC and This is DE. So we have to prove that These two are equal Okay, so again as we have been discussing so far Whenever there are two parts of a triangle to be proven to be equal The best way we know or one of the ways best ways we know is To prove that they are congruent parts of congruent triangles, right? We just need to identify the two triangles properly and as well as prove this Congruence establish the congruence between the two triangles Automatically the desired result comes. So let's find out which are the triangles of which are the triangles Whose part is BC and DE so BC clearly is a part of Triangle ABC. Is it it looks like the base of ABC and DE is part of two triangles one is DAE as well as DCE right But if you see on the left side of DE we have more information in terms of the angles being equal and other things But that's not the you know Not by any certainty we can see that only left-hand side of DE Whatever is there is going to be useful But looks like it's that's how Intuition is built up and hence we utilize those information So if we somehow prove that BAE this triangle this congruent to BAC then automatically their Basis in this case DE and BC will become equal So that's what we need to do how to prove that those two triangles are equal So some given information is going to be Helpful for us. So the two sides so this side AB of ABC look clearly and very closely ABC has this side AB and ADE has this side AD again now We have been given that AB is equal to AD so this becomes useful Similarly this part of AC of this triangle ABC This side is equal to this side of DE so I hope you're able to see these two triangles. These are the two triangles. I'll draw them here, this is a B See and our ADC is something like this this is AD and This goes to be So this is E and this is D. Okay, so this is a This is D and this is E So if you now look closely, this is equal to this Right and this side is equal to this side. It's already given AC is equal to a check and we have to just See whether these two angles are all equal Then our job is done. If you look closely guys It's given that You know, if I draw if I have to draw here. This was my AD is it it and Here if you draw This is This goes like that. So this is this goes to C in this diagram check so clearly this angle was given to be equal to this angle and If you look closely, this angle is common in both of them Right. So what I'm saying is check here This angle is equal to this angle already given and if I add this this one to first this one So I'll get This full Now this full is also equal to this full and hence our angle a in this case and a here will be equal Okay, I hope you got from this image. So let me now remove These parts so that it becomes easier for you to understand. Okay, so So now I am going to take away these things So the diagram will be a little clearer for you. Okay, so let me also take away this So these are not required any further. Just it was just for Understanding. Okay. So now let's begin the proof. So how to prove it? So I will say I will say angle a b d is equal to angle C a e given C a e this is given here Right, this is given The moment this is given. I can add one angle to both. What is that adkc? So I'm going to do this Look carefully a b d plus angle d ac. I'm going to add this angle to both sides d ac plus angle C a b Now the moment why do I do this? Because the moment I did that I got what a b d or a Sorry, this should not be a b d my bad. This should be b a d b a d D ad right so sorry for that. So this is d ad B a right now if you check d ad plus d ac is how much angle b a See and here d ac plus d a e D ac sorry plus C a e these are the two angles getting added. This will be angle b a Hope this is clear I d a e this is established now Let's come to Proving the congruence part. Okay. Now take triangle a or other b a C and triangle B a e these are the corresponding vertices in that order b ac b a e What do we know a Is equal to d a Isn't it? It's given It's given here right a b is equal to ad also a c a c is equal to a e given this is also given check here and We just prove from here. We can say Angle d ac is equal to angle b a e Okay, therefore conclusion therefore triangle b ac is Congruent to triangle D a e check the order of the vertices once again b is corresponding to be correct a corresponds to a correct C corresponds to be very good. So this is correct. Therefore Therefore, what can you conclude you can conclude that bc the corresponding sides must be equal So bc must be equal to Be so hence I can say dc is equal to Be Is it this is what we needed to establish and we got it Okay, so one thing is what is to be noted here is that you know There was one common element common angle which was added to make the two vertex angles equal, right? So this is you know keep this in mind So there will be many such cases where Two different angles this one and this one are equal and then the moment you add this common x these two triangles become equal as well That was one of the learning in this session Okay