Question

In a cyclic quadrilateral ABCD, it is given ∠A = (2x + 4)°, ∠B = (y + 3)°, ∠C = (2y + 10)° and ∠D = (4x – 5)°. Find the four angles.

Answer 1

The opposite angles of cyclic quadrilateral are supplementary, so
∠A +∠C = 180°
⇒ (2x + 4)° + (2y + 10)° = 180°
⇒ x + y = 83°
And
∠B + ∠D = 180°
⇒ (y + 3)° + (4x – 5)° = 180°
⇒ 4x+ y= 182°
Subtracting (i) from (ii), we have
3x = 99 ⇒ x = 33°
Now, substituting x = 33° in (i), we have
33° + y = 83° ⇒ y = 83° - 33° = 50°
Therefore
∠A = (2x + 4)° = (2 × 33 + 4)° = 70°
∠B = (y + 3)° = (50 + 3)° = 53°
∠C = (2y + 10)° = (2 × 50 + 10) °= 110°
∠D = (4x - 5)° = (4 × 33 - 5) ° = 132° – 5° = 127°
Hence, ∠A = 70°, ∠B = 53°, ∠C = 110° and ∠D = 127°.