Vertices A, B, C and D of a quadrilateral ABCD lie on a circle. ∠A is three times ∠C and ∠D is two times ∠B. What is the difference between the measures of ∠D and ∠C?
Explanation:
Given,
∠A is three times ∠C and ∠D is two times ∠B.
∠A = 3∠C .......(1)
∠D = 2∠B .......(2)
In a cyclic quadrilateral, opposite angles are supplementary.
∠A + ∠C = 180° and ∠B + ∠D = 180° ∠A + ∠C = 180°
3∠C + ∠C = 180° [From (1)]
4∠C = 180°
∠C = 45°
∠A = 3∠C = 135°
∠B + ∠D = 180°
∠B + 2∠B = 180° [From (2)]
3∠B = 180°
∠B = 60°
∠D = 2∠B = 120°
Difference between the measures of ∠D and ∠C = 120° - 45°
= 75°
Hence, the correct answer is Option C
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