 Hey, hi folks. How are you? I hope all of you are doing good and hail and hearty healthy sound enjoying life now We are going to take up another problem on congruent triangles and you must be cursing me that if someone is doing congruent triangles how can life be good but that's how it is right and I think questions are good and it gives a lot of mental exercise and We learn how to solve problems. So hence I like geometry proving problems. I hope if you're also Keen on doing geometries, then you have to solve more and more Problems right so what's the problem here? The question says AC is equal to BC. Okay. Where is AC? Okay? this is AC this is BC and DC a so DC a is equal to e CB e C Be I hope you are able to Wait a minute. Let me change this. Yeah, so what I'm saying is this angle here is These angles these are two equal angles clear Okay, and also angle this a is equal to angle B, which is also given Okay, so let me just remove those now We have to prove that triangle DBC. Where is DBC? Let's highlight D here can check D B C is Concurrent to e AC e AC, okay And hence obviously the moment they are congruent drops of other things can be found out We'll see DC is equal to e C and B D is equal to a you know that but I am going to complicate this problem once again and What I'm thinking of doing is Why don't we join B E? Again, and why don't we join D a as well? Okay now Can we prove this so over and above whatever is given can we prove that triangle a DB? a DB is congruent to triangle B a B a DB a DB B a that's one and hence can you prove that ad is Equal to ad is equal to be This is what is The extra thing you have to do. So let's do some mental exercise. Don't worry first. We'll prove what is given and then we'll try and go there Okay, so Let's go about the proof. What is to be proven? So as we have been Doing let's write given so that's given AC is equal to BC. No doubt about it. AC is equal to BC This is given and what else angle DC a DC a check DC a where is DC a DC a DC a yeah is equal to um ECB ECB yes right E C and B Very good and dbc angle dbc. Where is dbc? Let's check B D easier B is here C is here. So this angle is Equal to E a C. Yes is equal to angle E C folks, right Let me write it a little better way because Clarity of mind is what we are seeking. So E a C E write properly a C Correct. What do we need to prove prove that to prove? To prove what do we need to prove? Triangle what D? BC so D BC is congruent to triangle E a C E a C and for that What all do we need one thing you wouldn't have seen dbc, right? So dbc one thing is anyways given AC is equal to BC so that's given also One angle is also equal if you check see first is this side is equal to this side and This angle anyways is equal to this angle between the two triangles I hope you are able to identify the two triangles which talking about if not, let me draw this for you So this is the first triangle. This is the first triangle and The second triangle is this Right So just to highlight. Okay, and then I'll take it away just to highlight. So this and this Right these two triangles are To prove it. We have to prove that these two are equal Conquer it. Let me take away all of this. Otherwise, you'll get utterly confused. I Hope so. I'm taking away all this Yes, right. So So Yes, so now the picture the two triangles are clear in your mind So let's go about it. So in to prove in this triangle. So let's say in triangle DBC triangle DBC and Triangle EAC what all is known? What do we see? So one is clearly angle EAC or Deep angle DBC itself DBC itself is equal to angle EAC. So point number one One angle is established to be equal which one I'm talking about this one. Yeah, this is equal to this, right and AC is equal to or in this case the side will be BC. So right BC is equal to AC So I'm getting a So if you see this is the a Then this is the s. So we need one more a and we're done. What is that a? So either the top angles top angle meaning this D is equal to a e but that's what we have to prove actually see We have to visit. No, we don't need to prove that but anyways, we don't know whether D is equal to e But we know one more thing. What is that? Look at this. Let me write it here. So if you see angle a C e a C e is equal to Angle DCE or let me write e cd trace the points e cd e cd plus Plus, let me for the convenience sake. Let me write this angle as x Okay, and this angle also will be x because that's given so e cd plus x Okay, and angle Bcd Pcd is also angle e cd Plus x check both are true. Is it it? ECD happens to be the common angle and x you add to both you'll get to the two of these angles So hence from these two together I can say here It's a angle a So first I'll write Bcd because I have taken that triangle in the left-hand side. So angle B B Um cd right so Bcd is equal to cd this one will be equal to angle a c e and Ta-da, so hence the three so we get third a right So a so a s a criteria fulfilled, right? Then we can say that these two triangles are going to enhance dear friends triangle DBC is congruent to triangle E ac Right and the moment you say that Entire thing whatever yours is asked for is actually true. So DC where is DC? So DC and EC are the CPCT, right? So MCC DC and EC so you can write DC is equal to PC Come so this is established and BD is equal to AE. So where is BD? Check BD again This is also CT CPCT. So this BD and this EA will be equal So hence you'll write BD is equal to AE True both are true and reason is congruent parts of sorry corresponding parts of Now the evil mind what we'll do is we connect AD and BCB Do you think these two triangles are also congruent? this one ADB and BEA, can we do that? Yes, they are and let's look at these two triangles very carefully ADB and AEB Right these two triangles clearly AB happens to be the common side, right? And this angle here this one the EAB is equal to DBA it's also given equal and third we just proved that BD is equal to AE Isn't it? We just proved that isn't it? So hence Hence, let me now draw further. So you see triangle ADB mark the point. So ADB I'm taking first ADB I'll again draw it properly ADB triangle ADB ADB and triangle BEA Interesting these two right what will happen there clearly this AB is Equal to BA corresponding side C common sides So AB is equal to BA no doubt about it What else we also know that angle ABD is equal to angle BAE Who says I am not saying anything the question is saying right ABD Where is it? or Is it so? Yeah ABD, where was it given? Yeah, that is CBD is ABD actually so it's given here see ABD or ABD or CBD same thing, right? So and EAC EAC is equal to EAB. Yep. So hence these two are also equal Not else. We just proved that BD is equal to AE Right from where? from where from here Right BD is equal to AE. So this again if you see this is S A and S friends. So by SAS I can say I can say What can I say? I can say by SAS SAS congruence criteria. I can say again ABD triangle ABD Is congruent ABD ABD is equal to Triangle or whatever we had mentioned to you we should be mentioning the same this thing So let's not change the change the order of the points. So ADB had written. So let's write ADB only and This is congruent to triangle BEA BEA Perfect Right the moment ADB is congruent to BEA You don't need to do much here. So again AD will be equal to BE and that is CPCT And hence the extended version of the question is also Done and likewise if you want to really extend it further These are some food for thought for you. So you can prove DO is equal to OE OA is equal to OB all that right? So and yeah So these are you can establish also if this is M This is let's say this point is M. This point is N So you can check MCM will be equal to CN also, right? OM will be equal to ON so much so many symmetrical results You can establish and here is how questions are set. They will ask you prove that OM is equal to ON Correct, and that will be extended version of what Just solved which appeared to be very very simple. So these are food for thought try to prove OM is equal to OM. OD is equal to OE MA is equal to NB CM is equal to CN lots of lots of things you can do in one problem itself you can prove Lots of proving things. Okay, so I hope you understood what was the learning in this session Learning is we must be very very clear with the four rules of congruences Secondly, we must be very, you know, able to identify the right triangles where you can Find out equal parts, right? So identifying the right set of triangles also is an art It will happen not overnight. It will take some time So solve keep solving all such problems and you will see eventually you will be able to Find out exactly with two triangles to be found out and made congruent So that the given or the demand of the question can be that I hope you understood this So let's meet again next session new question. Okay, till then. Bye. Bye Take care