Question

In figure, ABCD is a cyclic quadrilateral. <CBQ = 48° and x = 2y. Find the value of y.

Answer 1

Given:

ABCD is a cyclic quadrilateral.

angle ∠CBQ = 48° and x = 2y.

In triangle BCQ, ∠DCB (an exterior angle) = ∠CBQ + y = 48° + y.

In triangle APB, ∠ABP = ∠CBQ = 48° because they are vertically opposite angles.

Hence, angle ∠DAB = ∠ABP + x = 48° + x.

Since ABCD is cyclic, opposite angles sum to 180°: ∠DCB + ∠DAB = 180°

Substitute angles: (48° + y) + (48° + x) = 180°

Group terms: 48° + y + 48° + x = 180°

96° + x + y = 180°

Rearrange: x + y = 180° − 96° 

= 84°

Given x = 2y, substitute: 2y + y = 84°

3y = 84°

y = `(84°)/3`

y = 28°