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Maths / Circles / Angle in a Semicircle
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Theorem 1  The angle in a semi- circle is a right angle.

Given     PQ is a diameter of a circle C(O,r) and dpi{120} large angle PRQ  is an angle in semi- circle.

To prove  large angle PRQ=90^{circ}

PROOF  We know that the angle subtended by an arc of a circle at its centre is twice the angle formed by the same arc at a point on the circle. So, we have

              large angle POQ=2angle PRQ

large Rightarrow ;;;180^{circ}=2angle PRQ                [large because POQ  is a straight line]

large Rightarrow ;;;angle PRQ=90^{circ}

 

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