Circle theorem proofs – GCSE Maths

Higher · Grade 8 · Non-calculator

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Proving a circle theorem is a short, named chain, usually from isosceles triangles (radii equal) and 'angle at the centre is twice…' or triangle-sum. Angle in a semicircle: the centre angle on the diameter is $180^{\circ }$, so the circumference angle on the same arc is $90^{\circ }$. Tangent-radius: the shortest distance from centre to tangent meets at $90^{\circ }$, or equal tangents plus radii make two congruent right-angled triangles.

Write a reason on every line. Draw radii you need, even if they were not printed. The proof is the marks — arriving at the named result without reasons is a show-that with no method.

The same theorems as the basics topic, but the marks are the reasons. Angle at the centre is twice the circumference; alternate segment; cyclic quadrilateral opposite angles sum to $180^{\circ }$; the tangent is perpendicular to the radius. Write the theorem in words next to each angle.

Objectives

• Use the centre-and-edge theorem with a reason.
• Use equal radii to get base angles.
• Write the reason in the answer, not only the number.

Key points

• Quote every circle theorem accurately in the standard wording.
• Spot isosceles triangles made by two radii and use their equal base angles.
• Write a proof as a numbered chain: statement, then reason, on every line.
• Prove a general result using letters such as $x$ and $y$, not just find one angle.

Worked examples

Questions and answers

GCSE Maths circle theorem proofs questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Doubling or halving an angle that is not on the same arc

   Check that the centre angle and the edge angle stand on the same arc first

2. Halving the centre angle to get a base angle of the radii triangle

   Use 180 minus the centre angle, then share the rest between two equal angles

3. Giving the angle with no reason

   Write the theorem in words next to the number, such as same arc or equal radii

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Set out the proof as a numbered list: statement on the left, reason on the right. Markers can then tick each line.
• In a general proof, define $x$ and $y$ at the very top; a proof that starts with an undefined letter loses marks.

Printable worksheet

The same questions are in a PDF you can print. Answers start on a later page.

Download worksheet (PDF)

Or use Print in your browser. Answers under each question are included when you print.

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More on Circle theorem proofs

• Circle theorem proofs worked examples and method
  The method in order, key facts and the full lesson explanation
• Circle theorem proofs practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.