In the given figure, O is the centre of the circle and ∠DAB = 50°. Calculate the values of x and y.

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Solution

O is the centre of the circle and ∠DAB = 50°.
OA = OB (Radii of a circle)
⇒ ∠OBA = ∠OAB = 50°
In ΔOAB, we have:
∠OAB + ∠OBA + ∠AOB = 180°
⇒ 50° + 50° +∠AOB = 180°
⇒ ∠AOB = (180° – 100°) = 80°
Since AOD is a straight line, we have:
∴ x = 180° – ∠AOB
= (180° – 80°) = 100°
i.e., x = 100°
The opposite angles of a cyclic quadrilateral are supplementary.
ABCD is a cyclic quadrilateral.
Thus, ∠DAB + ∠BCD = 180°
∠BCD = (180° – 50°) = 130°
∴ y = 130°
Hence, x = 100° and y = 130°

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