Angle Subtended by an Arc of a Circle

The angle made by two different equal arcs to the centre of the circle will also be equal.

There are two arcs in the circle AB and CD which are equal in length.

So \angle AOB = \angle COD.

Theorem 8: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at some other point on the remaining part of the circle.

In the above figure \angle POQ = 2\angle PRQ.

Theorem 9: Angles from a common chord which are on the same segment of a circle are always equal.

If there are two angles subtended from a chord to any point on the circle which are on the same segment of the circle then they will be equal.

\mathrm{\angle a=\left(\dfrac{1}{2}\right)\angle c} (By theorem 8)

 \mathrm{\angle b=\left(\dfrac{1}{2}\right)\angle c}

\mathrm{\angle a=\angle b}

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