# Write True or False and justify your answer in each of the following: If A, B, C, D are four points such that $\angle BAC = 30^{\circ} \;\; and \;\;\angle BDC = 60^{\circ}$, then D is the centre of the circle through A, B and C.

True

Solution:

Given, ∠BAC = 30° and ∠BDC = 60°

We know that,

The angle subtended at the centre by an arc is twice the angle subtended by it at any part of the circle.

Considering BC,

At centre, $\angle BDC = 60^{\circ}$

At any other point on the circle,

$\angle BAC = 30^{\circ}=\frac{1}{2} (60^{\circ})$

Hence the above rule is justified.

Therefore the given statement is true.

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