Question

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

Answer 1

Given an arc PQ of a circle subtending angles POQ at the centre O and PAQ at a point A on the remaining part of the circle. We need to prove that angle POQ = 2 angle PAQ.
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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