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

Question:

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

 

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 = 2ADC
⇒ 100° = 2ADC
∴ 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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