Question

In the figure, quadrilateral ABCD is cyclic. If m(arc BC) = 90° and ∠DBC = 55°, then find the measure of ∠BCD.

Answer 1

Given: m(arc BC) = 90°, ∠DBC = 55°

To find: ∠BCD

`∠BDC = 1/2 m(arc  BC)`   ...[Inscribed angle theorem]

∴ `∠BDC = 1/2 xx 90^circ`   ...[Given]

∴ ∠BDC = 45°   ...(i)

In ∆BCD,

∠BDC + ∠DBC + ∠BCD = 180°   ...[Sum of the measures of all angles of a triangle is 180°]

∴ 45° + 55° + ∠BCD = 180°   ...[From (i) and given]

∴ 100° + ∠BCD = 180°

∴ ∠BCD = 180° – 100°

∴ ∠BCD = 80°

∴ The measure of ∠BCD is 80°.