The circle theorems

2) Angles in the same segment

The angles at the circumference subtended by the same arc are equal. Put more simply, angles in the same segment are equal.

Circle with identical angles, a, at the circumference

Angle α = α.


Example

Calculate the angles p and q.

Circle with angles, 52degrees, 40degrees, p and q at the circumference

Angles in the same segment are equal. So p = 52° and q = 40°


Proof

Let the obtuse angle MOQ = 2x. Using the circle theorem, the angle at the centre is twice the angle at the circumference.

Circle with double angle (2x) at centre

Angle MNQ = x and angle MPQ = x.

Two angles (x) at circumference

Therefore angle MNQ = angle MPQ.


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