 Hello friends and how are you all doing today? My name is Priyanka. Let us discuss this question. It says the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Using above theorem, solve the following. In the figure O is the center of the circle angle O B A is given to us as 40 degrees and angle O C A is given to us as 30 degrees. We need to find the measure of angle B O C. Now this is a corresponding figure in which we need to find out the angle of B O C where we are given that angle O B A O B A is 40 degrees and O C A is 30 degrees. Right now first of all let us prove the given theorem. Here we need to prove that angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Right? Let us start with our solution. We are given that a circle with center O and arc A B subtended, sorry, subtending, subtending angle A O B at the center, angle A C B at any point let's say C on the remaining part of the circle. We need to prove that angle A O B, A O B is twice angle A C B. For that we need to first of all join C O, produce it to a point also. We have joined O A with O B. Now let's start with our proof. Now in triangle A O C, we can see that O A is equal to O C because both are the radii of a circle. Right? O A and O C. So therefore we can see that angle O C A, O C A that means this angle is equal to angle O A C because they are angles opposite to equal sides. Also, we have angle P O A is equal to angle O C A plus angle O A C as they are the exterior angle of a triangle and we know that the measure of exterior angle of a triangle is the sum of the opposite angles of a triangle. Since we know that angle O C A is equal to O A C. So we can say that therefore angle P O A is equal to twice any of these two angles because both of them are equal. So we can say that it's twice angle O C A. Being a simple reason as angle O C A is equal to angle O A C. Now in the same manner, we can say that angle P O P or P O P is equal to twice angle O C B. Let this be the first equation and this be the second equation. Now on adding one and two, we have angle P O A plus angle B O P and we can write angle B O P as P O B also is equal to twice. We have taken two common. We are left with these two angles or these two angles. So we have angle O C A plus angle O C B. That is equal to twice angle. This angle, angle A O P plus P O B is angle A O B, right? And O C A plus O C B is this whole angle. So we have angle C A C B, right? So hence we have proved the given theorem, right? This was the first part of the given question. The second part requires us to use this theorem that we have proved just now and solve the following. In figure O is the center of the circle. O is the center of the circle. This angle is given to us as 40 degree and this angle is given to us as 30 degrees. We need to find out the measure of angle B O C. Now, here we are given that O is the center of the circle. O B A is given to us as 40 degrees and angle O C A is given to us as 30 degrees. Further, we know that since O is the center of the circle, therefore O B will be equal to O A as they are the radii of the circle. So with this we can say that therefore angle O B A is equal to angle O A B and that is equal to this angle is given to us in the question itself as 40 degrees and we will write that angle opposite to equal size. Similarly, angle O C B since O C is equal to O A with the same reason, therefore angle O C A will be equal to angle O A C and that is equal to 30 degrees. As they are angles opposite to equal also. Now, we have this angle as 40 degrees and this angle as 30 degrees that we have found out above. So the whole angle B A C will be equal to the sum of these two angle that is angle O A B plus angle O A C. The values are 40 degrees and 30 degrees respectively. So this full angle will become 70 degrees and if this angle is 70 degrees then angle at the center will be twice this angle because this is the angle at the center subtended by the same arc B C. So therefore, we can write down angle B O C is equal to twice angle B A C because of the theorem proved above and it is equal to twice 70 degrees that is equal to 140 degrees. So the final answer to this session is angle B O C is 140 degrees. So this completes the session. Hope you understood the whole concept where I enjoyed it too. Have a nice day ahead.