# Proove that angle subtended by an arc on the center is double the angle made by it on circumference

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by shruti13

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by shruti13

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Let us join AO and extending it to point B.

Now, angle BOQ=angle OAQ+angle AQO ( Exterior Angle Property)

Also in triangle OAQ,

OA = OQ (Radii of the same circle)

angle OAQ = angle OQA (Angles opp. to equal sides are equal)

angle BOQ = 2 angle OAQ......................i

Similarly, angle BOP = 2 angle OAP................ii

From i and ii we get,

=> angle BOQ + angle BOP = 2(angle OAP+ angle OAQ)

=> angle POQ = 2 PAQ

Hence, Proved

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