Standard circle theorems
The 7 GCSE circle theorems are essential for Higher tier. Memorise the names AND the underlying reasons — exam questions demand both.
The 7 theorems
1. Angle at the centre = 2 × angle at the circumference
Subtended by the same arc/chord. The centre angle is double the circumference angle on the same side.
2. Angle in a semicircle = 90°
A triangle inscribed with one side as the diameter has a right angle at the third vertex.
3. Angles in the same segment are equal
Two angles subtended by the same chord, on the same side of it, are equal.
4. Opposite angles in a cyclic quadrilateral sum to 180°
A "cyclic quadrilateral" has all 4 vertices on a circle.
5. Tangent perpendicular to radius
At the point of contact, the tangent meets the radius at 90°.
6. Tangents from external point are equal
TA = TB when T is external and TA, TB are tangents.
7. Alternate segment theorem
The angle between a tangent and a chord equals the angle in the alternate segment (the angle subtended by the chord in the segment on the other side of the chord).
✦Worked example
A cyclic quadrilateral ABCD has ∠A = 95° and ∠B = 70°. Find ∠C and ∠D.
- ∠C = 180° − 95° = 85° (opposite cyclic angles).
- ∠D = 180° − 70° = 110° (opposite cyclic angles).
✦Worked example— Worked example — angle at centre
In a circle, ∠ACB = 50° at the circumference, subtended by chord AB. Find the angle AOB at the centre.
- ∠AOB = 2 × 50° = 100°.
✦Worked example— Worked example — alternate segment
A tangent at P touches a circle. A chord PQ is drawn. The angle between the tangent and PQ is 40°. Find the angle PRQ (R is on the circle, on the OTHER side of PQ).
- Alternate segment: ∠PRQ = 40°.
⚠Common mistakes
- Wrong side of the chord for "angles in the same segment" — both angles must be on the SAME side.
- Doubling the wrong angle — centre = 2 × circumference, not the other way.
- Cyclic quadrilateral confusion — all 4 vertices must be on the circle.
- Alternate segment misidentification — the angle in the OTHER segment, not the same one.
- Forgetting reasons — naming the theorem is required.
➜Try this— Quick check
In a cyclic quadrilateral, one angle is 105°. State the opposite angle.
- 180 − 105 = 75°.
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