G5: Congruence Criteria
Know and use the criteria for congruent triangles: SSS, SAS, ASA, RHS
📋 Key Concepts
Definition: Two shapes are congruent if they have the same shape and size - they can be exactly superimposed on each other.
Important: Congruent shapes have all corresponding sides and angles equal. The areas are also equal.
📝 The Four Congruence Criteria
SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
RHS (Right-angle-Hypotenuse-Side): If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle, the triangles are congruent.
📝 Using SSS
Example 1
Triangle ABC has sides AB = 5 cm, BC = 7 cm, AC = 4 cm. Triangle DEF has sides DE = 5 cm, EF = 7 cm, DF = 4 cm. Prove the triangles are congruent.
Solution:
AB = DE = 5 cm
BC = EF = 7 cm
AC = DF = 4 cm
All three corresponding sides are equal, so by SSS, triangles ABC and DEF are congruent.
📝 Using SAS
Example 2
In triangles PQR and XYZ: PQ = XY = 6 cm, QR = YZ = 8 cm, and angle PQR = angle XYZ = 45°. Prove the triangles are congruent.
Solution:
PQ = XY = 6 cm (first side)
Angle PQR = Angle XYZ = 45° (included angle)
QR = YZ = 8 cm (second side)
Two sides and the included angle are equal, so by SAS, triangles PQR and XYZ are congruent.
📝 Using ASA
Example 3
In triangles ABC and DEF: AB = DE = 5 cm, angle A = 50°, angle B = 60°, angle D = 50°, angle E = 60°. Prove congruence.
Solution:
Angle A = Angle D = 50°
AB = DE = 5 cm (included side)
Angle B = Angle E = 60°
Two angles and the included side are equal, so by ASA, triangles ABC and DEF are congruent.
📝 Using RHS
Example 4
Triangle ABC is right-angled at B. Triangle DEF is right-angled at E. AC = DF = 10 cm, AB = DE = 6 cm. Prove congruence.
Solution:
Both triangles are right-angled
AC = DF = 10 cm (hypotenuses are equal)
AB = DE = 6 cm (one pair of corresponding sides equal)
By RHS, triangles ABC and DEF are congruent.
📝 Non-Congruence Cases
AAA is NOT a congruence criterion: Knowing three angles only tells you the shape, not the size. The triangles could be similar but different sizes.
ASS is NOT a congruence criterion: The angle must be between the two sides (SAS), not just any angle.
Example 5
Two triangles have angles of 40°, 60°, 80°. Are they congruent?
Solution: Not necessarily. They are similar (same angles) but could have different sizes. We need information about the sides to prove congruence.
🧮 Worked Examples — Exam Style
Congruence = identical shape AND size. Four criteria: SSS, SAS, ASA (or AAS), RHS. AAA and SSA are NOT criteria.
Two triangles share: AB = DE = 6 cm, BC = EF = 8 cm, AC = DF = 10 cm. Congruent?
Yes — SSS (all three pairs equal)
B1 with criterion named. Naming the criterion IS the marks — a bare 'yes' does not score.
Why do two triangles with angles 30°, 60°, 90° but sides 3-4-5 and 6-8-10 fail congruence?
All angles match (AAA) but the sizes differ — they are similar, not congruent.
B1 for the similarity distinction. AAA fixes shape but not scale — the classic distractor.
Exam wording to expect: "prove/explain whether the triangles are congruent", name the criterion in every proof.
❓ Practice Questions
Q1: What does SSS stand for?
Q2: State the congruence criterion used when you know: two sides and the angle between them.
Q3: Why is AAA not a congruence criterion?
Q4: What does the H stand for in RHS?
Q5: Two right-angled triangles have hypotenuses of 13 cm and one side of 5 cm each. Are they congruent?
Q6: Triangles PQR and XYZ: PQ = XY = 7 cm, QR = YZ = 9 cm, angle PQR = angle XYZ = 50°. Are they congruent? Which criterion?
Q7: Two right-angled triangles have hypotenuse 13 and one leg 5 (each). Congruent? Criterion?
✅ Answers
- Side-Side-Side
- SAS (Side-Angle-Side)
- Three equal angles gives similar triangles (same shape), not congruent triangles (same size)
- Hypotenuse (the longest side of a right-angled triangle)
- Yes, by RHS criterion
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Side-Side-Side
Q2. SAS — two sides and the angle BETWEEN them
Q3. AAA gives similar triangles (same shape, any size)
Q4. Hypotenuse
Q5. Yes — RHS (right angle, hypotenuse, side)
Q6. Yes — SAS: two pairs of equal sides with the included angle equal
Q7. Yes — RHS (13, 5, and the right angle)
📌 Key Facts to Memorise
- SSS: three pairs of sides · SAS: two sides + INCLUDED angle
- ASA/AAS: one side + two angles · RHS: right angle + hypotenuse + side
- NOT criteria: AAA (shape only) · SSA (ambiguous — the angle not included)
- Congruent ⇒ mirror-image flips are allowed (reflections still congruent)
- In proofs: quote the criterion, then conclude corresponding parts equal (CPCTC)
🗺️ In the Exam
Congruence: 2–4 marks on Paper 1. The criterion name is the marks — the included-angle condition is what separates SAS from the SSA trap.
Links to: geometric proof (G6), similar shapes (G19), Pythagoras (G20).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always state the criterion you are using (SSS, SAS, ASA, or RHS)
- Make sure the angle in SAS is the included angle (between the two sides)
- Remember: AAA gives similarity, not congruence
- For RHS, both triangles must be right-angled
- Write a clear conclusion: "Therefore, triangles ABC and DEF are congruent by [criterion]"
🧠 Problem-Solving Strategies
Problem-Solving
The four congruence criteria are: SSS, SAS, ASA, AAS. RHS is a special case for right-angled triangles. Never use SSA (two sides and a non-included angle) — it is NOT a congruence criterion. Always state which criterion you are using and list corresponding parts clearly.
Multi-Step Problem
Triangle ABC has AB = 8 cm, BC = 6 cm and ∠ABC = 90°. Triangle PQR has PQ = 8 cm, QR = 6 cm and ∠PQR = 90°. Prove the triangles are congruent and find AC if PR = 10 cm.
Solution: Both triangles have two sides and the included right angle equal: AB = PQ, BC = QR, ∠ABC = ∠PQR = 90°. By SAS (or RHS), the triangles are congruent. Since they are congruent, AC = PR = 10 cm.
⚠️ Common Errors
Watch Out!
1. Wrong: Using SSA to prove congruence (two sides and a non-included angle) Correct: SSA is NOT a congruence criterion — it can produce two different triangles (the ambiguous case). Only use SSS, SAS, ASA, AAS, or RHS.
2. Wrong: Saying two triangles are congruent just because they have the same area Correct: Triangles with the same area can have completely different shapes. Congruence requires matching sides AND angles.
3. Wrong: For RHS, not checking that the angle is the right angle (not just any angle) Correct: RHS requires the Right angle, the Hypotenuse and one Side. The right angle must be explicitly stated or proven — it cannot be assumed.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: In the diagram, AB = CD, AD = BC, and AC is a shared diagonal. (a) Prove that triangle ABC is congruent to triangle CDA. (b) Hence show that AB is parallel to DC. (c) What type of quadrilateral is ABCD? Justify your answer fully.
(a) In triangles ABC and CDA: AB = CD (given), BC = AD (given), AC is common. By SSS, △ABC ≡ △CDA.
(b) Since the triangles are congruent, ∠BAC = ∠DCA (corresponding angles in congruent triangles). These are alternate angles formed by line AC crossing AB and DC. Since alternate angles are equal, AB is parallel to DC.
(c) ABCD is a parallelogram. We have shown AB ∥ DC. By a similar argument using the other pair of equal angles, AD ∥ BC. Both pairs of opposite sides are parallel, so it is a parallelogram.
Mark scheme: M1 identifying equal sides, A1 SSS congruence, M1 corresponding angles, A1 parallel lines reason, M1 other pair, A1 parallelogram justified
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Two triangles each have sides of 5 cm and 7 cm with an angle of 40°.
(a) If the 40° angle is between the two sides in both triangles, are they congruent?
(b) If the 40° angle is NOT the included angle in both triangles, must they still be congruent?
(c) A student claims "If two triangles have the same three angles, they must be congruent." Give a counterexample to disprove this.
Answers: (a) Yes — SAS congruence applies. Two sides and the included angle are equal. (b) Not necessarily — this is the SSA case which is ambiguous. Two different triangles can have the same two sides and a non-included angle. (c) Any two similar but differently-sized triangles are counterexamples. E.g. a triangle with angles 60°, 60°, 60° and sides 3, 3, 3 vs sides 5, 5, 5. Same angles but different sizes — they are similar, not congruent.
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