Vertices A, B, C and D of a quadrilateral ABCD lie on a circle. $$\angle$$A is three times $$\angle$$C and $$\angle$$D is two times $$\angle$$B. What is the difference between the measures of $$\angle$$D and $$\angle$$C?

Given,

$$\angle$$A is three times $$\angle$$C and $$\angle$$D is two times $$\angle$$B.

$$\angle$$A = 3$$\angle$$C.......(1)

$$\angle$$D = 2$$\angle$$B.......(2)

In a cyclic quadrilateral, opposite angles are supplementary.

$$\angle$$A + $$\angle$$C = 180$$^\circ$$ and $$\angle$$B + $$\angle$$D = 180$$^\circ$$

$$\angle$$A + $$\angle$$C = 180$$^\circ$$

3$$\angle$$C + $$\angle$$C = 180$$^\circ$$  [From (1)]

4$$\angle$$C = 180$$^\circ$$

$$\angle$$C = 45$$^\circ$$

$$\angle$$A = 3$$\angle$$C = 135$$^\circ$$

$$\angle$$B + $$\angle$$D = 180$$^\circ$$

$$\angle$$B + 2$$\angle$$B = 180$$^\circ$$  [From (2)]

3$$\angle$$B = 180$$^\circ$$

$$\angle$$B = 60$$^\circ$$

$$\angle$$D = 2$$\angle$$B = 120$$^\circ$$

Difference between the measures of $$\angle$$D and $$\angle$$C = 120$$^\circ$$ - 45$$^\circ$$

= 75$$^\circ$$

Hence, the correct answer is Option C