Circle theorems: alternate segment and multi-step problems

Harder circle questions chain several theorems together. The alternate segment theorem and isosceles triangles made by radii appear again and again.

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The alternate segment theorem

The angle between a tangent and a chord is equal to the angle in the alternate segment (the angle at the circumference on the other side of the chord).
If the tangent at A makes 64° with chord AB, then any angle ACB in the other segment is 64°.

Radii make isosceles triangles

OA = OB (radii), so triangle OAB is isosceles.
If angle OAB = 25°, then angle OBA = 25° and angle AOB = 180° − 50° = 130°.
The angle at the circumference on that arc is half: 65°.

Proving a theorem

To prove 'the angle at the centre is twice the angle at the circumference', join the point C on the circumference to the centre O and extend it.
This makes two isosceles triangles (radii). Each exterior angle at O is twice a base angle, and adding gives angle AOB = 2 × angle ACB.
In an exam, give a reason for every step, using the full wording of the theorem.
Worked example

PA and PB are tangents from P to a circle with centre O. Angle APB = 40°. C is on the major arc. Find angle ACB.

  1. Angles OAP and OBP are 90° (tangent ⟂ radius).
  2. Quadrilateral OAPB: angle AOB = 360° − 90° − 90° − 40° = 140°
  3. Angle at circumference = half: 70°

Answer: 70°

Key idea

Tangent and chord: the alternate segment theorem. Two radii: an isosceles triangle. Chain the facts and give a reason for each one.

Key words
Segment
The region of a circle between a chord and an arc.
Alternate segment theorem
The angle between a tangent and a chord equals the angle in the alternate segment.
Chord
A straight line joining two points on a circle.
Arc
Part of the circumference of a circle.
Exterior angle
The angle between one side of a polygon and the extension of the next side. The exterior angles of any polygon add to 360°.

All GCSE and International GCSE Mathematics key terms and definitions →

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  1. The tangent at A makes an angle of 36° with the chord AB. C is on the circle in the alternate segment. Find angle ACB.

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    Answer: 36 °

    By the alternate segment theorem, angle ACB = 36°.

  2. O is the centre. A and B are on the circle and angle OAB = 18°. Find angle AOB.

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    Answer: 144 °

    OA = OB, so angle OBA = 18°. Angle AOB = 180° − 2 × 18° = 144°.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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