 Hello friends welcome again to another session on gems of geometry, so We have come with another theorem in this theorem. It says in a triangle the angle bisector of the smaller angle is greater than the angle bisector of the Greater angle. Okay, that means let me try and explain it to you all so let us say in triangle ABC, so Here is the given part. There's a trying given triangle ABC given Triangle ABC. Okay, such that angle B Angle B is less than angle C This is given. Okay. We have to prove To prove What do we need to prove we need to prove so angle B so it says in a triangle the angle bisector of The smaller angle so BM is bisector of Bisector of angle B here and what else CN is by sector of Angle C Okay Now since angle B is less than angle C. So we have to prove that BM is Greater than CN Okay, so angle bisector of the smaller angle that is B is greater than the angle bisector of the greater angle Let's try and prove this guy then for this proof We will be using the theorem we proved in the previous session and let me write it here on The sidelines so we have to use this theorem theorem is that ah chord subtending subtending Smaller angle Smaller angle smaller angle on the circle is Smaller Okay, what does this mean so you can go back and check our previous session so What I meant was this let's say AB is a smaller chord and let's say CD is a larger chord so hence AB will subtend let's say a PB and CD is subtending C QD so We can say if if angle a P B is less than angle C QD C QD then AP AB, sorry AB is smaller than CD This we proved last time Okay in the last session so you will be using this particular Theorem so you can go back to the previous session and see what exactly is the theorem about now coming back to this proof So hence we are saying we have to prove VM is greater than CN for that We need to do some construction. What's the construction? So construction is Pick a point here and let's say M dash Okay, such that B M dash. Let me just join this M. That's C. Yeah, so construction Angle M dash C N Is equal to angle N BM N B M or MB M dash whichever right so we have to choose a point M dash in such a way that M dash C N Is equal to N BM, right? So what is NBM if you if you really see this is nothing but this is half of angle B Is it it because BM is the angle by sector similarly here this angle will be also half of angle B Because we are purposefully choosing this point now if you look closely if you do this construction, what will happen? so proof since of since Angle M dash C N is equal to angle N BM dash So you can imagine this to be a God Let's say I'm joining M dash N Okay, so let me join it a little bit more clearly science. Okay, let me join this Right, so if you see it it looks like What does it look like guys? It looks like it looks like that there is a circle which is passing through these points And then so passing through these point and hence hence it looks like it looks like B N M dash and C are consiclic Or consiclic that is they lie on the same Circle because if you see M M dash N looks like a chord. So this is a chord which is subtending Equal angle at two different points of the circle Correct. So they are consiclic if they are consiclic then what do we know? Yes, so now let's so these are the consiclic points. So hence if you see Angle N B C Right is equal to nothing but two times half of angle B Correct, right. Can I say that so hence this can be written as half of angle B plus half of angle B Now one of these angle half of angle B Can be written as let's say angle M dash C in right plus half of angle B right now This can be written as angle M dash C N because M dash C N was also half of angle B So hence, can I not now instead of this equality? Can I not? Write this that it is less than half of angle C Why? Because angle B is less than angle C given so angle B by 2 will be less than angle C by 2 So hence I can write this Right, so hence it is less than angle M dash C N plus half of angle C is nothing but angle N C B This angle here is half of angle C Because N C was the bisector Correct, so this means this is less than right. What is it less than M dash C B? M dash C B so you can see this right because M dash C N Plus N C B is M dash C B correct, so what did we Conclude B now infer that in angle NBC is less than angle M dash C B now if you see these are two angles on the same circle NBC NBC can you see NBC here NBC is let me just highlight this angle NBC. This is NBC guys, right? and what is The other one is this this angle right so NBC is less than M dash C B That means the chord Which is subtending NBC will be lesser than the chord which is some subtending M dash C B So what is and and how how do we know that because of this particular theorem so hence? We can conclude from here that is since NBC is less than so NBC is opposite NC so NC NC is less than B M dash Isn't it NC will be less than B M dash because N M dash C B are concitalic points and M dash being the chord And similarly NC being the chord and BM is also BM dash is also the chord so hence angle subtended by B N will be I'm sorry angle subtended by NC so NC you can see Angle subtended by NC is nothing but angle B and angle subtended by B M dash is angle C Not not angle C M dash C B rather, right? So this is from this theorem. We conclude that NC is less than B M dash Now BM dash is part of BM. So hence from this we can conclude that NC is Less than B M. Also. Is it B M dash itself is less than B M. So hence NC will be less than B M Let me just make it in the same frame. So you can understand. So I hope now this is better. So NC is less than B M we prove that and that is what we needed to prove isn't it? So if you see To prove was here this one. So hence this is from here directly B M Is greater than C N correct So hence we conclude that in a triangle the angle by sector of the smaller angle is Greater than the angle by sector of the greater angle