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Geometric Proof

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on geometric proof helps you revise constructing geometric proofs. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. State the reason for every step using known angle and shape facts.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Geometric proof requires you to justify each step of an argument using known geometrical facts, rather than relying on how a diagram looks.

Diagrams are usually not drawn to scale, so measuring is never acceptable. Every statement must follow from a theorem you can name — angles on a straight line, angles in a triangle, properties of parallel lines, congruence conditions or circle theorems.

The structure matters as much as the content. Set out one statement per line, each with its reason, building towards the required conclusion. A proof that reaches the right answer without stated reasons scores very little.

Revision notes

Structuring a proof

Write one statement per line, each followed by its justification.

For example: angle \(ABC = 70^\circ\) because angles on a straight line add to \(180^\circ\). Build each step from the previous ones until you reach the conclusion.

Using the right reasons

Quote theorems by their standard names, such as alternate angles are equal or the angle at the centre is twice the angle at the circumference.

Vague phrases like it looks equal or by the diagram earn nothing.

Proving congruence or similarity

For congruence, state each matching pair with its reason, then name the condition: SSS, SAS, ASA or RHS.

For similarity, show that the corresponding angles are equal, which is sufficient for triangles.

Key points

  • Every step needs a stated reason.
  • Never measure from the diagram.
  • Diagrams are not drawn to scale.
  • Quote theorems by their standard names.
  • Set out one statement per line.
  • Build logically towards the conclusion.

Worked examples

Example 1

Give the reason why two angles on a straight line sum to \(180^\circ\).

Working

\[\text{A straight line is a half turn}\]consider the geometry
\[\text{Angles on a straight line add to } 180^\circ\]state the standard reason

Example 2

What must you state to prove two triangles congruent?

Working

\[\text{Each matching pair with its reason}\]list the equal sides and angles
\[\text{Then name the condition, such as SAS}\]the condition completes the proof

Example 3

Why is measuring the diagram not acceptable in a proof?

Working

\[\text{Diagrams are not drawn to scale}\]the drawing may be misleading
\[\text{Each step must follow from a theorem}\]proof requires logical justification

Common mistakes

  • Relying on how the diagram looks.

    Diagrams are not to scale, so appearance proves nothing.

  • Giving the answer without reasons.

    The reasons carry most of the marks in a proof question.

  • Using vague justifications.

    Quote the theorem by name rather than saying it is obvious.

  • Assuming what you are trying to prove.

    Each step must follow from something already established.

Exam tips

  • Write one statement per line with its reason alongside.
  • Learn the standard wording of the common theorems.
  • Mark angles on the diagram as you establish them.
  • Finish with a clear concluding statement.

Key terms

Proof
A logical argument justifying a result.
Reason
The theorem justifying a step.
Congruent
Identical in shape and size.
Theorem
An established geometrical fact.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.