 Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says, let the vertex of an angle A, B, C be located outside a circle and let the sides of the angle intersect equal chords A, D and C, E with the circle. Prove that angle A, B, C is equal to half the difference of the angle subtended by the chords A, C and D, E at the center. So let us see the solution to this question. We see that this is the given circle, then A, D and C, E extended meet at B. That means this lies outside the circle. Then we see that the sides of angle intersect equal chords A, D and C with the circle. Now what we have to prove in the question is that A, B, C is equal to half the difference of the angle subtended by the chords A, C and D, E at the center. So first of all we write down mathematically what we have to prove. We have to prove that angle A, B, C that means this angle is equal to half of angle A or C minus angle D or E. That means these are the angles subtended by A, C and D, E at the center. So we have to prove that angle A, B, C is equal to half of angle A or C minus angle D or E. So let us see the proof to this question. Since an exterior angle of a triangle is equal to sum of interior opposite angles, so by this we have that in triangle B, D, C that means this triangle we will have angle A, D, C that means this angle is equal to angle D, B, C plus angle D, C, B that means this angle is equal to this angle plus this angle by this exterior angle theorem. So we write down what we have just seen. We see that in triangle B, D, C angle A, D, C is equal to angle D, B, C plus angle D, C, B. We name this one since angle at the center is twice the angle at a point the remaining part circle at angle A, D, C that means this angle, angle A, D, C is equal to half of angle A or C and for the same reason we have angle D, C, B that means this angle is half of angle D or E. So we have angle A, D, C is equal to half of angle A or C and angle D, C, B is equal to half of angle D or E. Now what we do we put the values of angle A, D, C and angle D, C, B in equation one. So we get, we name this two, we have this one. So from one and two of angle A or C is equal to angle D, B, C we have because we see that angle D, B, C is same as same angle A, B, C. D is equal to angle A, B, C, B for angle D or E on the other side and we have angle A, B, C is equal to half of angle A or C minus angle D or E. Good question and enjoy the session. Have a good day.