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