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Circle theorems and angle reasons · Higher notes

Higher · Geometry · AQA references G10

Circle theorems connect radii, chords, tangents and angles. Identify what the angle is standing on before applying a theorem: angles subtended by the same chord in the same segment are equal, and the centre angle is twice the corresponding circumference angle. A tangent meets its radius at a right angle. Combine these facts with ordinary angle sums and state a reason for each step. A sketch helps you track the correct arc; a diagram that looks convincing is not itself a proof.

Learning objectives

  • Choose a circle theorem and give a reason for an angle calculation.
  • Use radii and tangents to solve a connected angle problem.

Key ideas

The centre angle is twice the circumference angle subtended by the same arc.

θcentre=2θcircumference\theta_{\text{centre}}=2\theta_{\text{circumference}}

Angles in the same segment are equal; an angle subtended by a diameter is a right angle.

θsemicircle=90∘\theta_{\text{semicircle}}=90^\circ

Opposite angles of a cyclic quadrilateral sum to a straight angle.

A+C=180∘A+C=180^\circ

A tangent is perpendicular to its radius. Tangents from the same external point have equal lengths.

The tangent-chord angle equals the angle subtended by that chord in the opposite segment.

Worked examples

Example 1

Tangents from T touch at A and B; O is the centre. The angle at T is 56 degrees. Find the non-reflex centre angle.

A radius meets its tangent at a right angle.

∠OAT=∠OBT=90∘\angle OAT=\angle OBT=90^\circ
∠ATB=56∘\angle ATB=56^\circ

Angles in the quadrilateral sum to a full turn.

∠AOB=360∘−90∘−90∘−56∘\angle AOB=360^\circ-90^\circ-90^\circ-56^\circ

Answer

∠AOB=124∘\angle AOB=124^\circ

Example 2

A centre angle is 130 degrees. Find the circumference angle subtended by the same arc.

θcircumference=130∘2\theta_{\text{circumference}}=\frac{130^\circ}{2}

Answer

65∘65^\circ

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Find an unknown angle using circle theorems.

AO2 · Reasoning

Give a linked chain of angle reasons or prove a stated angle relationship.

AO3 · Problem solving

Combine circle theorems with algebra, triangles or parallel-line facts.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Match the correct arc or chord before choosing a theorem. An angle in a semicircle is a right angle, not a straight angle.

Try it yourself

Attempt each question before opening its answer.

Check 1: A centre angle is 130°. Find the circumference angle subtended by the same arc.

Answer: 65°

The circumference angle is half the corresponding centre angle.

Check 2: ABCD is cyclic. Angle A = 68°. Find opposite angle C.

Answer: 112°

Opposite angles in a cyclic quadrilateral sum to 180°.

Explain it in your own words

  • Name three circle theorems and draw a small example of each.
  • Explain why a tangent and its radius form a right angle.
  • Write a chain of angle reasons rather than only a final angle.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Geometry specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.