Circle Theorems

Visual summary

What you will learn

• Use the angle at the centre theorem.
• Use angles in the same segment.
• Recognise cyclic quadrilateral angle facts.
• Use tangent-radius and alternate segment theorems.
• Use the angle in a semicircle theorem.
• Write reasons clearly for proof-style questions.

Key facts before you start

Why this matters

Circle theorem questions reward accurate reasons. A correct angle without a theorem statement can lose marks, especially in proof questions where the examiner needs to see why each step follows.

Prior knowledge

You should already be comfortable with:

• Angle facts.
• Triangles and quadrilaterals.
• Parallel-line reasoning.
• Basic circle vocabulary.

Clear explanation

Main idea

Circle theorems are angle facts that work because points, chords, radii and tangents sit in fixed relationships. Start by naming the centre, tangent point, chord or cyclic quadrilateral before doing arithmetic.

Method

The angle at the centre is twice the angle at the circumference standing on the same arc. Angles in the same segment are equal. Opposite angles in a cyclic quadrilateral add to 180°. The angle in a semicircle is 90° because it stands on a diameter.

Answer tip

A tangent is perpendicular to the radius at the point of contact. The alternate segment theorem links a tangent-chord angle to the angle in the opposite segment. Write the theorem reason beside each angle step, not just at the end.

If a diagram has several circle facts, work one angle at a time. Mark the arc, chord, diameter, tangent point or cyclic quadrilateral you are using, then write a short reason such as "angle at centre is twice angle at circumference" or "opposite angles in a cyclic quadrilateral add to 180°". If two angles look equal but do not stand on the same chord or arc, do not use the same-segment theorem.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For circle theorems, marks usually come from identifying the correct theorem, showing the angle calculation and writing a precise reason such as angle at the centre, angles in the same segment, cyclic quadrilateral, angle in a semicircle, tangent-radius or alternate segment theorem. Strong answers also name the relevant arc, chord, diameter, tangent point or opposite angles so the reason matches the diagram.

Common mistakes

• Using a theorem on the wrong arc: check that the centre and circumference angles stand on the same chord or arc.
• Forgetting theorem reasons: a correct number may not get full marks if the proof asks you to explain why.
• Mixing up cyclic facts: opposite angles add to 180°, while angles in the same segment are equal.
• Missing the tangent point: tangent-radius is 90° only at the exact point where the radius meets the tangent.
• Assuming equal-looking angles are equal: same-segment angles need the same chord or arc, not just a similar position in the drawing.

Extension challenge

A cyclic quadrilateral has one angle of 112 degrees. Find the opposite angle and name the circle theorem used.

Exam-board guidance

Circle theorems appear within GCSE Maths geometry content, usually with stronger extension emphasis. Exact wording and theorem combinations vary, but pupils should expect angle chasing, proof-style reasoning, tangent-chord recognition and clear theorem names.

Next lesson

Next, continue with Sine Rule, Cosine Rule and Area of a Triangle.