 Hello and welcome to the session. The given question says, in figure 2, PQ is the diameter of the circle with center O if angle PQR is 65 degrees, angle SPR is 40 degrees, angle PQM is 50 degrees, find angle QPR, angle PRS, angle QPN. Now for the sake of simplicity let us name some of these angles, let angle SRP be angle 1, angle PRQ be angle 2, let angle RPQ be angle 3, angle QPN be angle 4, angle QMP be angle 5. Now first let us write down what we are given. Here we are given first that PQ is a diameter of a circle with center O and also we are given some of the angles, angle PQR is equal to 65 degrees, angle SPR is equal to 40 degrees and angle PQM is equal to 50 degrees and we have to find first angle PR, then we have to find angle PRS and lastly we have to find angle QPN. Let us now start with the proof. Now here we are given that PQ is a diameter, so this implies angle 2 is equal to angle 5 is equal to 90 degrees. Since angle in a semicircle is of measure 90 degrees and here PQ is a diameter so it divides the circle into two equal parts and angle PMQ that is angle 5 and angle PRQ that is angle 2 lie on the semicircle therefore each is of measure 90 degrees. Let this be equation number 1. Now let us consider triangle PQR angle 2 plus angle 3 plus angle PQR is equal to 180 degrees since sum of all the angles of a triangle is 180 degrees. Now angle 2 is of measure 90 degree plus angle 3 plus angle PQR is equal to 180 degrees and angle PQR is given to us as 65 degrees therefore angle 3 is equal to 180 degree minus 90 degree plus 65 degrees and this implies 180 degree minus 150 degrees and this gives 25 degrees therefore measure of angle 3 is equal to 25 degrees. Let this be equation number 2. So this implies that angle PQR is equal to 25 degrees. Now let us consider triangle PMQ here again sum of three angles of this triangle is 180 degrees so we have angle 4 plus angle 5 plus angle PQM is equal to 180 degrees. Now angle 5 is equal to 90 degrees since angle in the semicircle is of measure 90 degree so in place of angle 5 we will put 90 degrees plus angle PQM is of measure 50 degrees and angle 4 we shall be finding out. So this is equal to 180 degrees so this implies that angle 4 is equal to 180 degrees minus 140 degrees which is equal to 40 degrees so measure of angle 4 is 40 degrees this implies that angle 2 PM is equal to 40 degrees and now we know that in a cyclic correlator the sum of opposite angles is of measure 180 degree so in cyclic quadrilateral SRQP here angle SPQ plus its opposite angle is SRQ is equal to 180 degrees since sum of opposite angles of a cyclic quadrilateral 180 degrees. Now angle SPQ angle 3 and SRQ can be written as angle 1 plus angle 2 is equal to 180 degrees now let us substitute the values angle SPR is 40 degrees angle 3 is 25 degrees plus angle 1 as it is angle 2 is angle in a semicircle so it is of measure 90 degree this is equal to 180 degrees so this implies that angle 1 is equal to 180 degrees minus 40 degrees plus 25 degrees plus 90 degrees so this is equal to 180 degrees minus on adding we get 155 degrees and this is equal to 25 degrees so angle SRP is 25 degrees therefore angle SRP is equal to 25 degrees hence our answer is angle PR is equal to 25 degrees angle PRS is equal to 25 degrees and angle QPM is equal to 40 degrees so this completes the session hope you understood it bye and take care