 Welcome again guys in another problem-solving session in this question It's been given that there is a quadrilateral ABCD a when bo are the bisectors of angle a and b so before doing anything Let's first draw a quadrilateral. So let's say this is a quadrilateral. Okay, let me name it as a b c D a b cd is a quadrilateral A when bo are the bisectors of angle a and b. Okay, so let us now try to Draw bisectors, so let's say this is A o and this one should be bo. Okay, so let this point be oh Okay, so when it is a bisector so we can easily name it as x So this will also be x This will be why and this will be why because a when bo are Angle bisectors of angle a and b respectively. You have to prove that Aob So this angle is a ob isn't it a ob is equal to c plus d by 2 okay, so At the very first step The moment we see that the desired result is to find sum of two angles c and d So let me just mark these angles with a different color. Let's say this is c. This is d and this is c So we are talking about some of these two angles now Whenever there are Cases where some of angles are involved then was first thing which comes to mind is the angle some property So let's try and evaluate from that perspective from that angle. Let's see whether we are able to solve this question now Whenever Angle some property if a quadrilateral is concerned more often than not angle some property of triangle is also involved So you must be very very clear with that Concept so let's say in this in this case Let's start from angle a ob itself before that we have to do some rituals and the rituals is Rituals are you have to just write all this so what is given? given is abcd is a quadrilateral is a quadri Lateral now many times people are a verse of writing all this, but then this is a very integral part of communicating mathematics in communicating in mathematics, right? So hence you have to write given in geometric cases, especially you have to write given and you have to also mention what is to be Proved and then you proceed with your proof. So abcd is a quadrilateral. It's given then a o bisects so you can simplify and write bisects angle a and Bo bisects angle b Okay, what is to prove so to prove? to prove to prove angle a ob so keep in mind where is angle a ob so this is angle a ob is it angle a ob is Equal to angle c plus angle d upon 2 also keep in mind This is angle d this angle c also keep in mind where our angle c and d so you know where it is So this is c this angle is c and this angle is d Okay, now let's proceed further. So So what is angle a ob guys? So what do we know about a ob? Let's write that first. So a ob plus x plus y is equal to 180 degrees. Is it it why because this is angle some property of a triangle Okay, a ob is equal to x plus y is equal to 180 degrees now. What do we do? So since you know You will proceed with this one step at a time. So x a ob has been or a ob is there in my First equation now. We need to you know, just find out x and y in terms of the desired Result that is c and d. What is x? x is nothing but angle a y2 because A is being bisected by o. Okay, similarly y is angle b by 2 isn't it So hence equation one. So let me let it be equation one so equation one from one you can say from one angle a ob is equal to 180 degrees minus angle a y2 minus angle b by 2 right why because 180 degree minus x minus y is nothing but minus a by 2 minus b by 2 Okay, very good. Now another relationship with a b and c and d. What do we know? So angle a plus angle b Plus angle c plus angle d is equal to 360 degrees This is known. This is angle some property of a quadrilateral Right, this is what we know already That means I can say angle a plus angle b is equal to 360 degrees Minus angle c minus angle d. I took angle c and d on the other side This implies I can say So basically we got a plus b isn't it? So hence half of angle a Plus angle b and why am I doing this? You'll get to know just now Half into 360 degrees Minus angle c minus angle d isn't it? Just multiplying the both sides of the equation by half. Okay So hence if you simplify this you'll get 180 degrees minus c or minus c By 2 and minus angle d by 2 Isn't it so half angle a plus b is equal to this Now let's say this equation was 2 and this one is 3 Now you simply deploy 3 in 2 Right. So what will it become? So angle a OB now you can write here from 2 and 3 Angle a OB. So this is angle a OB is equal to 180 degrees and Minus a by 2 minus b by 2 can be written as angle a By 2 plus angle b by 2 the minus sign is common Which is nothing but 180 degrees minus half Times angle a plus angle v Correct. I took two common one by two common. Now. What is this if you see this particular? Value we have already calculated and you can use this here So half angle a plus b is equal to this So let's make use of this particular relationship. So hence what do we get I get? This is nothing but 180 degrees minus half angle a plus b can be written as Minus 180 degrees minus angle c by 2 minus angle D by 2 why because this item here is equal to this here Right. Why because you check this one is same as this one, isn't it? So hence what is the final result? So it is 180 degrees minus 180 degrees Plus angle c by 2 plus angle d by 2 isn't it just multiplying by minus sign this goes off So hence you can say it is c plus angle d by 2 and if you notice this is what we wanted to Find out angle c plus angle d by 2 is angle a o b. So again So angle a o b is angle c plus angle d by 2 and hence I write hence Proof, okay, so what did we learn we learn that? this question was totally around Anglesome property of triangle and a quadrilateral so hence you start with one point and then try to relate to That particular starting point to final desired result and we used in between the two concepts that is Anglesome property of a triangle and anglesome property of a quadrilateral. I hope you understood the solution part. Thank you