Question

ABCD is a cyclic quadrilateral. M (arc ABC) = 230°. Find ∠ABC, ∠CDA, and ∠CBE.

Answer 1

M(arc ABC) = 230°

M(arc ADC) = 360° - M(arc ABC) … complete circle is 360°

M(arc ADC) = 360° - 230° = 130°

∴ ∠AOC = 130°

The angle subtended by an arc at the point on a circle is equal to half of the angle subtended by the same arc at the center

Here arc ADC subtends ∠AOC at center and ∠ABC on a circle

∴ ∠ABC = (1/2) × ∠AOC

= 1/2 × 130°

= 65°

∴ ∠ABC = 65°

∠ABC + ∠CBE = 180° …linear pair of angles

∴ 65° + ∠CBE = 180°

∴ ∠CBE = 180° - 65° = 115°

∴ ∠CBE = 115°

∠CDA + ∠ABC = 180° …opposite pair of cyclic quadrilateral ABCD

∴ ∠CDA + 65° = 180°

∴ ∠CDA = 180° - 65° = 115°

∴ ∠CDA = 115°

Hence ∠ABC = 65°, ∠CDA = 115° and ∠CBE = 115°