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GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 4–8 · Geometry

Congruent Triangles and Geometric Proof

Two shapes are congruent if they are exactly the same shape and size. Edexcel tests this in two ways: naming the right condition (SSS, SAS, ASA, RHS) and writing a short congruence proof. The proof is a 3–4 mark question that rewards a clear, reasoned layout.

1What congruent means

Congruent shapes are identical in shape and size. One can be moved onto the other by a rotation, reflection or translation. Similar shapes have the same shape but can be a different size.

Key ideaCongruent: all corresponding sides and angles are equal.
Similar: corresponding angles are equal and sides are in the same ratio.

2The four conditions

To prove two triangles are congruent, you only need to show one of these:

SSS: three pairs of equal sides
SAS: two sides and the angle between them
ASA (or AAS): two angles and a corresponding side
RHS: right angle, hypotenuse and one other side
These don’t work
  • AAA doesn’t prove congruence. Three equal angles only shows the triangles are similar.
  • SSA (two sides and an angle that is not between them) doesn’t prove congruence, except for the right-angled case RHS.

3How to write a congruence proof

  1. Name the two triangles, with corresponding vertices in the same order.
  2. State three facts, each with a reason (given, common side, vertically opposite angles, property of a square, …).
  3. Name the condition (SSS, SAS, ASA or RHS) and conclude that the triangles are congruent.
Worked example
ABCD is a parallelogram. Prove that triangles ABC and CDA are congruent.
  1. AB = CD (opposite sides of a parallelogram are equal).
  2. BC = DA (opposite sides of a parallelogram are equal).
  3. AC = CA (common side).
  4. So triangle ABC ≅ triangle CDA (SSS).
Congruent by SSS
Exam tipThe reasons earn the marks. “AB = CD” on its own scores less than “AB = CD because opposite sides of a parallelogram are equal”.

4Using congruence to prove other results

Once two triangles are congruent, all their corresponding parts are equal. That lets you prove facts about angles and lengths.

Worked example
Triangle ABC is isosceles with AB = AC. M is the midpoint of BC. Prove that angle ABC = angle ACB.
  1. AB = AC (given).
  2. BM = CM (M is the midpoint of BC).
  3. AM = AM (common side).
  4. So triangles ABM and ACM are congruent (SSS).
  5. Therefore angle ABM = angle ACM, that is, angle ABC = angle ACB (corresponding angles of congruent triangles).
Base angles of an isosceles triangle are equal
Worked example
Lines AB and CD cross at M, which is the midpoint of both lines. Prove that triangles AMC and BMD are congruent.
  1. AM = BM (M is the midpoint of AB).
  2. CM = DM (M is the midpoint of CD).
  3. Angle AMC = angle BMD (vertically opposite angles).
  4. So triangle AMC ≅ triangle BMD (SAS).
Congruent by SAS
Common mistakes
  • Using AAA or SSA as a congruence condition.
  • Using SAS when the angle is not between the two sides.
  • Stating facts without reasons.
  • Writing the vertices in a mismatched order, e.g. ABC ≅ DFE when A ↔ D, B ↔ E.
  • Confusing congruent (same size) with similar (scaled).

★Key Vocabulary Recap

Congruent (≅)
Identical in shape and size.
Included angle
The angle between two given sides.
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Common side
A side shared by both triangles.
Specification reference (Pearson Edexcel GCSE (9–1) Mathematics 1MA1: G5, G6, G7)
  • G5: use the basic congruence criteria for triangles (SSS, SAS, ASA, RHS).
  • G6: apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results; construct simple proofs.
  • G7: identify congruent and similar shapes.
Next stepReady to practise? Try the 10 exam-style questions, then check your working against the worked solutions.
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Practice questions

GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 4–8 · Geometry

Congruent Triangles and Geometric Proof — Practice Questions

10 exam-style questions (24 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Foundation and Higher · Grades 4–5
  1. (1 mark)
    Grade 4
    Two triangles have sides 5 cm, 7 cm, 9 cm and 9 cm, 5 cm, 7 cm. Are they congruent? Give the condition.
  2. (1 mark)
    Grade 4
    Two triangles each have angles 40°, 60° and 80°. Explain why this is not enough to show they are congruent.
  3. (1 mark)
    Grade 5
    Triangle P has sides 6 cm and 8 cm with an angle of 50° between them. Triangle Q has sides 6 cm and 8 cm with an angle of 50° between them. Which condition shows they are congruent?
  4. (1 mark)
    Grade 5
    Two right-angled triangles both have a hypotenuse of 13 cm and one other side of 5 cm. Are they congruent? Give the condition.
  5. (2 marks)
    Grade 5
    Triangle ABC ≅ triangle DEF, with A ↔ D, B ↔ E and C ↔ F. Angle ABC = 72°, AB = 9 cm and EF = 11 cm. Write down angle DEF, the length DE and the length BC.
Higher · Grades 6–8
  1. (3 marks)
    Grade 6
    ABCD is a parallelogram. Prove that triangle ABC is congruent to triangle CDA.
  2. (3 marks)
    Grade 6
    Lines AB and CD intersect at M, the midpoint of both. Prove that triangles AMC and BMD are congruent.
  3. (4 marks)
    Grade 7
    ABCD is a kite with AB = AD and CB = CD. Prove that angle ABC = angle ADC.
  4. (4 marks)
    Grade 7
    ABCD is a square. E lies on BC and F lies on CD, with BE = CF. Prove that triangles ABE and BCF are congruent.
  5. (4 marks)
    Grade 8
    Triangle ABC is isosceles with AB = AC. D is the point on BC such that AD is perpendicular to BC. Prove that D is the midpoint of BC.
    HintWhich condition uses a right angle and a hypotenuse?
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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Worked solutions

Show worked solutions
GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 4–8 · Geometry

Congruent Triangles and Geometric Proof — Worked Solutions

Full worked solutions to all 10 questions (24 marks), set out the way an Edexcel examiner wants to see them.

Foundation and Higher · Grades 4–5
  1. (1 mark)
    Grade 4
    Two triangles have sides 5 cm, 7 cm, 9 cm and 9 cm, 5 cm, 7 cm. Are they congruent? Give the condition.
    Solution
    1. All three pairs of sides are equal.
    Yes, SSS
  2. (1 mark)
    Grade 4
    Two triangles each have angles 40°, 60° and 80°. Explain why this is not enough to show they are congruent.
    Solution
    1. Equal angles only show they are the same shape (similar). One could be an enlargement of the other.
    AAA shows similarity, not congruence
  3. (1 mark)
    Grade 5
    Triangle P has sides 6 cm and 8 cm with an angle of 50° between them. Triangle Q has sides 6 cm and 8 cm with an angle of 50° between them. Which condition shows they are congruent?
    Solution
    1. Two sides and the included angle.
    SAS
  4. (1 mark)
    Grade 5
    Two right-angled triangles both have a hypotenuse of 13 cm and one other side of 5 cm. Are they congruent? Give the condition.
    Solution
    1. Right angle, hypotenuse and one side are equal.
    Yes, RHS
  5. (2 marks)
    Grade 5
    Triangle ABC ≅ triangle DEF, with A ↔ D, B ↔ E and C ↔ F. Angle ABC = 72°, AB = 9 cm and EF = 11 cm. Write down angle DEF, the length DE and the length BC.
    Solution
    1. Corresponding parts of congruent triangles are equal.
    2. DEF = ABC = 72°; DE = AB = 9 cm; BC = EF = 11 cm.
    72°, 9 cm, 11 cm
Higher · Grades 6–8
  1. (3 marks)
    Grade 6
    ABCD is a parallelogram. Prove that triangle ABC is congruent to triangle CDA.
    Solution
    1. AB = CD and BC = DA (opposite sides of a parallelogram are equal).
    2. AC is common.
    3. So ABC ≅ CDA (SSS).
    SSS
  2. (3 marks)
    Grade 6
    Lines AB and CD intersect at M, the midpoint of both. Prove that triangles AMC and BMD are congruent.
    Solution
    1. AM = BM and CM = DM (M is the midpoint of both lines).
    2. Angle AMC = angle BMD (vertically opposite).
    3. So AMC ≅ BMD (SAS).
    SAS
  3. (4 marks)
    Grade 7
    ABCD is a kite with AB = AD and CB = CD. Prove that angle ABC = angle ADC.
    Solution
    1. In triangles ABC and ADC: AB = AD (given), CB = CD (given), AC is common.
    2. So triangle ABC ≅ triangle ADC (SSS).
    3. Therefore angle ABC = angle ADC (corresponding angles of congruent triangles).
    Proved via SSS
  4. (4 marks)
    Grade 7
    ABCD is a square. E lies on BC and F lies on CD, with BE = CF. Prove that triangles ABE and BCF are congruent.
    Solution
    1. AB = BC (sides of a square are equal).
    2. Angle ABE = angle BCF = 90° (angles of a square).
    3. BE = CF (given).
    4. So triangle ABE ≅ triangle BCF (SAS).
    SAS
  5. (4 marks)
    Grade 8
    Triangle ABC is isosceles with AB = AC. D is the point on BC such that AD is perpendicular to BC. Prove that D is the midpoint of BC.
    Solution
    1. In triangles ABD and ACD: angle ADB = angle ADC = 90° (AD ⟂ BC).
    2. AB = AC (given): these are the hypotenuses.
    3. AD is common.
    4. So ABD ≅ ACD (RHS), hence BD = CD and D is the midpoint of BC.
    Proved via RHS
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