 Hello friends, welcome again to another session on triangles. So in the given question, besides AB and AC of a triangle, ABC are produced to P and Q. So you can see AB is produced to P and this is Q and it's bisectors of PBC. So PBC, so PO is the bisector and QCB bisector is CO and and you have to prove that angle BOC, that means this angle is equal to 90 degrees minus half angle A. So let's say this is X and let's say this is Y and this, yeah, so instead of naming it as with letters, let's name them with numbers. So if this is one, so let's say this angle, if this is two, then clearly this angle is two, isn't it? And if this is three, so this angle also happens to be three, right? Why? Because OB is the bisector of angle PBC and OC is the bisector of BCQ. So I hope this figure is clear and let me call this angle as four and let's call this angle as five and this angle as six, right? So so many numbers and angles here, but never mind, we'll be able to prove it. Now we have to prove BOC, right? So what do we know about BOC? It is part of triangle BOC, isn't it? So we can say, let's first write the, you know, whatever is the mandatory steps, usually we write these things, right? Given is, what is given? OB bisects, OB bisects, OB bisects, angle PBO, no, sorry, angle PBC and OC bisects angle BCQ or QCB, right? This is given and we have to prove, to prove, what do we need to prove? We need to prove, angle BOC is equal to, angle BOC is equal to 90 degrees minus half angle A. Let's try and prove this, how to prove this? So as I told you, we can start from the triangle BCO, isn't it? So triangle in triangle, in triangle BCO, BCO, what can we, what do we see? We see that, angle one plus angle, two plus angle three is 180 degrees, no doubt about it. Why? Because this is called ASP, angle sum, angle sum property, angle sum property of a triangle, isn't it? Very clear, no problem. Now, three is half of angle PBC. So I can say angle one, or first of all, let's write this, that angle two times angle three, two times angle three is equal to angle four plus angle C, six, right? And why is that? Two times angle three, if you check, this is two times angle three, and this will be equal to interior opposite angle, sum of interior opposite angle, which happens to be four and six. So that's what I have written over here. And let's write, this is because of exterior, exterior angle property, exterior angle theorem, correct? We learned this in the previous sessions. So that means two angle, three is angle four plus six. That means angle three is equal to half angle four plus angle six. Let's name the equations also. So one, this is two. Okay, then third, let me write it here in this part so that the figure is also in the same frame. So I can also say similarly, for the other pair, similarly, I can say two times angle two, which is the external angle, which one, this one will be equal to this plus this, isn't it? Interior opposite. So two times angle two is angle four plus angle five. That means angle two is equal to half angle four plus angle five. I hope this is clear, right? Now what? We know that angle four plus five plus six itself is 180 degrees. Or let's write, let's start from here, from one. So can I not say from one, from one, from one? What do we say? We say angle one is equal to 180 degrees minus what? Angle two plus angle three. Just reordering or, you know, taking angle two and angle three on the RHS. Now 180 degrees. Now from two and three, if you see, I can replace angle two by what? You check this relationship and I am replacing angle two in this here by half angle four and angle five. And then angle three is here. So I can replace this by half angle four plus angle six, isn't it? Let's simplify this one. So this is nothing but 180 degrees minus half angle four, half angle four plus half angle five plus half angle four again plus half angle six. Correct? That means 180 degrees minus angle four because half plus half. If you add these two, you'll get half plus half is one. So angle four plus half angle five plus angle six, isn't it? Now if you look closely, four, five and six, what is four, five and six? So you can see angle four plus angle five plus angle six is 180 degrees clearly. Let us say this to be four or rather I can say angle five plus angle six is 180 degrees minus angle four. Let me call it equation number four. And if you go back to this equation, so what I'm going to do is let me just copy this and take it to... So let me copy this and let me take this to the place where we are solving so that it becomes clear to everybody in the same frame. So hence I'm writing this also. Now if you see, if I take this one now, if I take this, what can I say? And what was this? Where did we all start? This is all angle one, isn't it? All are angle one. So I can say now angle one is equal to 180 degrees minus angle four and half angle five plus six. So you see this can be written as half angle five plus angle six. Don't you agree that this is 90 degrees minus half angle four? How? By dividing the entire equation by two, this by two, this by two. Is it it? So half angle five plus six will be 90 minus half angle four. So let's replace that. So we had half angle five plus six here, which we were writing here. So 180 minus angle four and then this can be written as 90 minus half angle four. So if you simplify, this will become angle one is equal to 180 degrees minus angle four. If you add these two, one minus half is half. So half angle four comes out and then minus 90 degrees. Now if you add these two, you will get 90 degrees minus half angle four, correct? And what was angle one? Let's check. Angle one was BoC and what was angle four? Angle A. So hence, can I not replace this by angle BoC is equal to 90 degrees minus half angle A. And this is what we needed to prove and we have proved it. Correct? So angle BoC is equal to 90 degree minus half angle A. This is what this theorem or question demanded, right? So we could prove this. What all did we apply? What underlying concept learning? What are the learnings learnings are? Some angles, some property we deployed and exterior angle theorem we deployed and using these theorems, we could establish this result.