 Welcome friends to another problem-solving session in this question. It's given that OQ bisects angle A or C and OP bisects C O B then you have to prove that Q OP is a right angle Okay, so OQ bisects this thing that means this is X Then this is also X Bisection angle bisector means it divides the angle into two equal parts. Okay Now similarly since OP is a bisector of C O B as well, so this will be Y This will be Y Okay now what can we Say so you have to start like this solution should always be like that So you are writing solution you will be saying first of all what is given? given is OQ bisects Angle A O C and OP bisects Angle B O C now you would be thinking that it is a you know Double work if it's already given why do you need to write given but this is how you know This is a standard procedure to you know solve geometry problems it also gives you a clear understanding of the problem and You don't miss out on critical information. Okay, so the critical information here was OQ bisects A O C and OP bisects B O C. Okay, so you have to prove What do you need to prove so that gives you the clear-cut picture of the objective you have to prove that angle Q OP is equal to 90 degrees. This is what is the Objective, so how do you prove it? So let's start the proof. So you do it now We will use the given information OQ bisects A O C. So hence you can label them as X and X and Y and Y so you can say X 2 X is equal to Angle A O C Is it it and similarly 2 Y is equal to angle B O C Right now what you can say angle A O C plus angle B O C is 180 degrees This is again a critical step why 180 degrees and within brackets you can say they are they are forming linear pair, isn't it? So O C is the ray standing on line AB. So hence linear pair. So you can write. This is 1 This is 2 so hence from 1 and 2 and 3 No, but I have not mentioned 3 never mind. This is 3 So from 1 plus 2 1 and 2 and 3 you can write 2 X plus 2 Y is equal to 180 degrees Okay, so that means twice of X plus Y is equal to 180 degrees That means X plus Y is 90 degrees Oh, wow, wonderful. So X plus Y C. What are what is X? This was X But this was also X and this was Y and this was also Y So X plus Y either you take this and this or you take this and this So clearly the second case is helpful for me. So X plus Y. Can I not write X as Q O C? so this is angle Q O C and Y as C O B right so angle C O P and this is equal to 90 degrees now Q O C and C O P put together Q O C. So this is Q O C and C O P put together is nothing but Q O P, right? So Q O C and C O P put together is angle Q O P which is 90 degree hence Proved this is what our objective was Okay, so in this case we use the knowledge of linear pair and the concept of angle bisectors, right? So this is the learning of this problem