Question

In a cyclic quadrilateral ABCD, ∠A : ∠C = 3 : 1 and ∠B : ∠D = 1 : 5; find each angle of the quadrilateral.

Answer 1

 Let ∠A and ∠C be 3x and x respectively

In cyclic quadrilateral ABCD,

∠A + ∠C = 180°

(Pairs of opposite angles in a cyclic quadrilateral are supplementary)

`=>` 3x + x = 180°

`=> x = (180^circ)/4 = 45^circ`

∴ ∠A = 135° and ∠C = 45°

Let the measure of ∠B and ∠D be y and 5y respectively

In cyclic quadrilateral ABCD,

∠B + ∠D = 180°

(Pair of opposite angles in a cyclic quadrilateral are supplementary are supplementary)

`=>` y + 5y = 180°  

`=> y = (180^circ)/6 = 30^circ`

∴ ∠B = 30° and ∠D = 150°