Angles in the same segment - Higher

The angles at the circumference subtended by the same arc are equal.

More simply, angles in the same segment are equal.

Angles a = a

Circle with identical angles, a, at the circumference

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.

p = 52^\circ

q = 40^\circ

Proof

Let the obtuse angle MOQ = 2x.

Circle with angle at the centre, labelled 2x.

Using the circle theorem, the angle at the centre is twice the angle at the circumference.

Angle MNQ = x and angle MPQ = x.

Circle with angle at the centre, labelled 2x and two angles at the circumference labelled x.

Therefore angle MNQ = angle MPQ.