 Hello and welcome to the session. In this session we discussed the following question which says in the given figure AB is the chord of a circle with center O and AB is produced to C such that BC is equal to OB. Also CO is joined and produced to meet the circle in D if angle ACD is equal to Y degree and angle AOD is equal to X degree proves that X is equal to 3Y. First we shall recall some facts which says that angles opposite to equal size of a triangle are equal. In a triangle the exterior angle is equal to the sum of the interior opposite angles. This is the key idea for this question. Now we move on to the solution. This is the figure given to us. We are given AB is the chord of the circle then BC is equal to OB. CO is joined and produced to meet the circle in point D and we are given angle ACG is equal to Y degrees and angle AOD is equal to X degrees and we need to prove X is equal to 3Y. Consider the triangle OBC in this we are given OB is equal to BC so this means that angle BOC is equal to angle BCO. Now since angle BCO is equal to Y degrees so both these angles would be equal to Y degrees and the reason for this is that the angles opposite equal sides in a triangle are equal. Now in triangle OBC the sides OB and BC are equal so the opposite angles would also be equal. Now for triangle OBC angle OBA is the exterior angle and we know that inner triangle exterior angle is equal to the sum of the interior opposite angles so angle OBA would be equal to the sum of interior opposite angles that is angle BOC plus angle BCO. Now angle BOC and BCO are each equal to Y degrees so this would be equal to Y degree plus Y degree equal to 2Y degree. So we have now got angle OBA equal to 2Y degree. Next we consider the triangle OBA in this we have OA is equal to OB as they are the radar of the circle so this means that the angles opposite these two equal sides would also be equal that is angle OBA is equal to angle OAB. Now that we have got OBA as 2Y degrees so we get OBA equal to OAB equal to 2Y degrees. Next we have for triangle AOC angle AOD is the exterior angle and so this would be equal to the sum of the interior opposite angles of the triangle AOC. So angle AOD is equal to angle OAC plus angle OCA. Now we are given that angle OCA is Y degrees and angle OAC or you can say OAB is equal to 2Y degree so this is equal to 2Y degree plus Y degree equal to 3Y degree. Now we were given that angle AOD is equal to X degrees so we have X degrees is equal to 3Y degrees so this means that X is equal to 3Y. We were supposed to prove this so hence proved this complete C session hope you have understood the solution for this question.