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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.
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Solution
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.
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