In the given figure, O is the centre of the given circle and measure of arc ABC is 100°. Determine ∠ADC and ∠ABC.

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Solution

We know that the angle subtended by an arc is twice the angle subtended by it on the circumference in the alternate segment.
Thus, ∠AOC = 2∠ADC
⇒ 100° = 2∠ADC
∴ ∠ADC = 50°
The opposite angles of a cyclic quadrilateral are supplementary and ABCD is a cyclic quadrilateral.
Thus, ∠ADC +∠ABC = 180°
⇒ 50° + ∠ABC = 180°
⇒ ∠ABC = (180° – 50°) = 130°
∴ ∠ADC = 50° and ∠ABC = 130°

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In the figure, O is the centre of the circle and the measure of arc ABC is 100^o. The measure of angle ABC will be

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