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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.
728519_43630f26da354046b81fd1d89cbf5e18.png


Solution

Given that:
$$\angle DAE=70^{\circ}$$
$$(i)\angle BCD$$
$$\angle BAD=180^{\circ}-\angle DAE$$    (By Linear Pair)
$$=180^\circ-70^\circ=110^\circ$$
$$\angle BCD+\angle BAD=180^\circ$$   (By opposite angles properties of a cyclic quardilateral)
$$\Rightarrow \angle BCD=180^\circ-110^\circ=70^\circ$$
$$(ii)\angle BOD=2\angle BCD=2\times 70^\circ=140^\circ$$    (Angle subtended by an arc at the centre is double of the angle subtended at the circumference.)
$$(iii)\angle OBD=\cfrac12(180^\circ-\angle BOD)$$   (By properties of  triangle )
$$=\cfrac12(180^\circ-140^\circ)$$
$$=20^\circ$$

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