 Hello and welcome to the session I am Deepika here. Let's discuss the question which says prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle. Now we know that the length of tangents drawn from an external point to a circle are equal. So let's start the solution. A given a circle with center O is the sides a, b, b, c, d and dA of a quadrilateral a, b, c, d as respectively. By a, b of a quadrilateral a, b, c, d subtend equal to 180 degree. At the opposite sides of a quadrilateral circumscribing a circle subtend on its opposite side b, c, subtend angle b, o, c, angle b, o, c is equal to 180 degree. Construction, join, o, p, o, q, o, r and o, s. Triangle, a, o, s, triangle a, o, p, we have is equal to o, p, radii of the same circle, oa is equal to oa common. Again as is equal to ap because length of tangents drawn from an external point to a circle are equal. Therefore triangle a, o, s is congruent to triangle a, o, p by s, s, s congruency condition. Therefore angle a, o, s let us take this as angle 1 is equal to angle a, o, p let us take this as angle 2. Similarly we can say angle 3 is equal to angle 4, angle 5 is equal to angle 6 and angle 7 is equal to angle 8. Angle 1 plus angle 2 plus angle 3 plus angle 4 plus angle 5 plus angle 6 plus angle 7 plus angle 8 is equal to 360 degree. Angles at a point, angle 1 is equal to angle 2. So we can write angle 1 plus angle 2 as twice angle 1 to angle 4. So we can write angle 3 plus angle 4 as twice angle 3 and twice angle 4. Again angle 5 is equal to angle 6. So we can write angle 5 plus angle 6 is equal to twice angle 5 and twice angle 6 and angle 7 is equal to angle 8. So we can write angle 7 plus angle 8 is equal to twice angle 7 and twice angle 8 plus angle 7 is equal to 360 degree and angle 1 plus angle, this implies angle 3 plus angle 6 plus angle 7 is equal to 180 degree and angle 1 plus angle 4 plus angle 5 plus angle 8 is equal to 180 degree. Again we can write this as plus angle 3 plus angle 6 plus angle 7 is equal to 180 degree and angle 1 plus angle 8 plus angle 4 5 is equal to 180 degree. Now in the figure angle 2 plus angle 3 is angle A O B plus angle 7 is angle A O B. So this implies equal to 180 degree because angle 2 plus angle 3 is equal to angle A O B, 6 plus angle 7 is equal to angle C O D, angle 1 plus angle 8 is angle A O D, 5 is angle B O C. We have proved that the opposite sides of a quadrilateral circumscribing is circle supplementary angles at the center of the circle that is angle A O B plus angle C O D is equal to 180 degree and angle A O D plus angle B O C is equal to 180 degree. Hands proved. I hope the solution is clear to you. Bye and take care.