Question

In the figure given, O is the centre of the circle. $\angle$DAE$={70}^o$. Find giving suitable reasons, the measure of.
(I) $\angle$BCD
(II) $\angle$BOD
(III) $\angle$OBD.

Answer 1

Given that:

$\angle DAE={70}^{\circ }$

$\left(i\right)\angle BCD$

$\angle BAD={180}^{\circ }-\angle DAE$ (By Linear Pair)

$={180}^{\circ }-{70}^{\circ }={110}^{\circ }$

$\angle BCD+\angle BAD={180}^{\circ }$ (By opposite angles properties of a cyclic quardilateral)

$\Rightarrow \angle BCD={180}^{\circ }-{110}^{\circ }={70}^{\circ }$

$\left(ii\right)\angle BOD=2\angle BCD=2\times {70}^{\circ }={140}^{\circ }$ (Angle subtended by an arc at the centre is double of the angle subtended at the circumference.)

$\left(iii\right)\angle OBD=\frac{1}{2}({180}^{\circ }-\angle BOD)$ (By properties of triangle )

$=\frac{1}{2}({180}^{\circ }-{140}^{\circ })$

$={20}^{\circ }$