24)In the figure ‘O’ is the centre of the circle and measure of AOC is
1000
.Find ADC and ABC. (Give reason).
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Step-by-step explanation:
Angle subtended by an arc is twice the angle subtended by it on the circumference in the alternate segment.
Here arc ABC makes ∠ AOC = 100° at the center of the circle and ∠ ADC on the circumference of the circle.
∴ ∠ AOC = 2∠ ACD
⇒ ∠ ACD = ½ (∠AOC)
⇒ = ½ × 100° [∠ AOC = 100°]
⇒ ∠ ACD = 50°
The opposite angles of a cyclic quadrilateral are supplementary.
ABCD is a cyclic quadrilateral and thus,
∠ ACD + ∠ ABC = 180°
= 180° – 50° [∵ ∠ ADC = 50°]
= 130°
∴ ∠ ABC = 130°
∴ ∠ ACD = 50° and ∠ ABC = 130°
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