Circle theorems - Higher - OCRAngles in the same segment - Higher

Circles have different angle properties described by different circle theorems. Circle theorems are used in geometric proofs and to calculate angles.

Part ofMathsGeometry and measure

Angles in the same segment - Higher

The angles at the by the same 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 , 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.

Cyclic quadrilaterals - Higher

Angles at the centre and circumference - Higher

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