Question

The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)°, ∠C = (x + y)°, ∠D = (3y – 10)°. Find x and y, and hence the values of the four angles. 

Answer 1

We know that, by property of cyclic quadrilateral,

Sum of opposite angles = 180°

∠A + ∠C = (6x + 10)° + (x + y)° = 180°   ......[ ∵ ∠A = (6x + 10)°, ∠C = (x + y)°, given]

⇒ 7x + y = 170°   ......(i)

And ∠B + ∠D = (5x)° + (3y – 10)° = 180°   .....[∵ ∠B = (5x)°, ∠D = (3y – 10)°, given]

⇒ 5x + 3y = 190°

On multiplying equation (i) by 3 and then subtracting equation (ii) from them, we get

3 × (7x + y) – (5x + 3y) = 510° – 190°

⇒ 21x + 3y – 5x – 3y = 320°

⇒ 16x = 320°

⇒ x = 20°

On putting x = 20° in equation (i), we get

7 × 20 + y = 170°

⇒ y = 170° – 140°

⇒ y = 30°

∴ ∠A = (6x + 10)°

= 6 × 20° + 10°

= 120° + 10°

= 130°

∠B = (5x)°

= 5 × 20°

= 100°

∠C = (x + y)°

= 20° + 30°

= 50°

∠D = (3y – 10)°

= 3 × 30° – 10°

= 90° – 10°

= 80°

Hence, the required values of x and y are 20° and 30°, respectively and the values of the four angles i.e., ∠A, ∠B, ∠C and ∠D are 130°, 100°, 50° and 80°, respectively.