AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 15

3 marks · Medium difficulty · Reasoning

Determine whether two triangles are always, sometimes, or never congruent given conditions on their side lengths, angles, or enlargement scale factor.

Practise this question

Question

Question 15 asks to tick the correct box (Always congruent, Sometimes congruent, Never congruent) for three statements comparing two triangles A and B: 'A's sides are the same lengths as B's sides', 'A's angles are the same as B's angles', and 'A is an enlargement of B with scale factor 2'. Worth 3 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 15 shows 3 marks total: B3 for all three correct, B2 for two correct, B1 for one correct. The correct options are: 'A's sides are the same lengths as B's sides' is Always congruent; 'A's angles are the same as B's angles' is Sometimes congruent; 'A is an enlargement of B with scale factor 2' is Never congruent.

How to answer it

Understanding Congruence vs Similarity in Triangles

📋 What This Question Tests

This question assesses your conceptual understanding of congruence versus similarity for 2D shapes (specifically triangles):

  • Congruent shapes: Exactly the same shape and exactly the same size (corresponding sides and angles are identical).
  • Similar shapes: The same shape (identical angles), but can be different sizes (scaled by an enlargement scale factor).
  • Conditions for triangle congruence, notably the SSS condition.

Question 15: Triangle Statements

Deciding whether triangles A and B are Always, Sometimes, or Never Congruent [3 marks]

✅ Correct Tick Table

Statement Always congruent Sometimes congruent Never congruent
A's sides are the same lengths as B's sides ✔
A's angles are the same as B's angles ✔
A is an enlargement of B with scale factor 2 ✔
Mark Allocation:
• B3: All 3 rows correct.
• B2: Any 2 rows correct.
• B1: Any 1 row correct.

💡 Statement 1: Same Side Lengths

Answer: Always congruent

  • If two triangles have corresponding sides of exactly the same length, they satisfy the SSS (Side-Side-Side) congruence criterion.
  • Triangle side lengths rigidly lock the angles into place. It is impossible to make two different shaped triangles with identical side lengths.

📐 Statement 2: Same Angles

Answer: Sometimes congruent

  • Having three identical angles (AAA) guarantees that the triangles are similar, but not necessarily congruent.
  • Why sometimes?
    • If the scale factor between them is 1, they have equal sides and are congruent.
    • If one triangle is enlarged (e.g. scale factor 2), they have identical angles but different sizes, so they are not congruent.

❌ Statement 3: Enlargement Scale Factor 2

Answer: Never congruent

  • A scale factor of 2 means every side of triangle A is twice as long as the matching side of triangle B ( length A = 2 × length B ).
  • For shapes to be congruent, corresponding sides must have equal lengths (scale factor must be exactly 1).
  • Since the side lengths are strictly different, they can never be congruent.

Examiner Tips & Guidance

🧠 Exam Technique & Common Pitfalls

  • AAA does not mean Congruence: A common misconception is thinking "all angles the same" means shapes must be congruent. Think of an equilateral triangle with sides 2 cm and another with sides 10 cm; both have angles 60°, 60°, 60°, but one is much bigger!
  • Mark scheme rules for tick boxes:
    • Only place one tick per row. If a row contains more than one tick, the mark for that row is lost.
    • If you change your mind, clearly cross out your old answer. The mark scheme states: "if a tick and a cross are used in the same row, mark the tick."

💡 The 4 Triangle Congruence Conditions

Commit these four criteria to memory for GCSE:

  • SSS: Three pairs of equal sides.
  • SAS: Two sides and the included angle equal.
  • ASA / AAS: Two angles and a corresponding side equal.
  • RHS: Right-angle, Hypotenuse, and one other Side equal.

Topics

Geometry and measures · 3.4.1 Properties and constructions · 3.4.2 Mensuration and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.