Prove two triangles are congruent by matching sides and angles and quoting SSS, SAS, ASA or RHS. It is Higher only, aimed at grade 7. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Two triangles are congruent if they are the same size and shape. The legal conditions are SSS, SAS, ASA (or AAS) and RHS (right angle, hypotenuse, another side). AAA is similar, not congruent. SSA is not a legal condition (ambiguous). Write the three matching facts and the condition, with vertices in corresponding order: △ABC≡△FDE.
A proof question is a short chain: given equal sides or a shared side, plus a parallel-line angle reason, then SAS (say). Shared sides are a gift — write 'common side'. Do not assume extra equal angles from a sketch.
SSS, SAS, ASA (or AAS), RHS. AAA is similar, not congruent. SAS is two sides and the included angle — the angle must sit between the two sides. Write which vertices match.
Objectives
Recognise SAS and ASA conditions.
Recognise AAS when two angles and a non-included side are given.
Recognise SSS and RHS conditions.
Key points
State the four congruence conditions and what each letter means.
Pick out matching sides and angles from a diagram, with a reason for each.
Write a proof as three matching facts, each with a reason, then the condition.
Use properties of shapes (parallelogram, kite, isosceles triangle, circle) to justify equal sides and angles.
Worked examples
Example 1
Triangle ABC has two sides and the included angle known. Name the condition.
2 marks
Identify the given geometric parts: two side lengths and one angle.
Check the position of the angle: it is between the two known sides, so it is the included angle.
State the congruence condition: SAS.
Answer: SAS
Common slips
Slip: Assume three sides are given. Only two sides are known.
Slip: Assume the angle is opposite one of the sides. The problem explicitly states the angle is included.
Slip: State ASA. Two sides and the included angle is SAS.
Example 2
Triangle ABC has two angles and the included side known. Name the condition.
2 marks
Identify the given parts: two angles and one side length.
Check the position of the side: it lies between the two angles, so it is included.
State the congruence condition: ASA.
Answer: ASA
Common slips
Slip: Assume two sides and one angle are given. Two angles are specified.
Slip: Assume the side is non-included. The side connects the two angle vertices.
Slip: State AAS. When the side is included between two angles, the criterion is ASA.
Example 3
Triangle ABC has two angles and a side that is not between them. Name the condition.
3 marks
Identify the given parts: two angles and one side.
Check the position of the side: it is not between the two known angles.
State the congruence condition: AAS.
Answer: AAS
Common slips
Slip: Assume three angles are given. A side length is included in the information.
Slip: Assume the side is between the angles. The question specifies the side is not between them.
Slip: State ASA. Two angles and a non-included side is AAS.
Questions and answers
GCSE Maths congruent triangles questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A triangle has two sides and the included angle. Name the condition.
Show answer
Answer: SAS
The angle sits between the two known sides.
Criterion is SAS.
M1 — The angle sits between the two known sides.
A1 — Criterion is SAS.
Question 2 Easy2 marks
A triangle has two sides and the included angle on a second drawing. Name the condition.
Show answer
Answer: SAS
Angle is between the two sides.
Criterion is SAS.
M1 — Angle is between the two sides.
A1 — Criterion is SAS.
Question 3 Medium2 marks
Triangles ABC and DEF have AB = DE and BC = EF. To prove congruence by SAS, which angle in DEF must equal angle B?
Show answer
Answer: angle E
The angle must sit between the two known sides.
The required angle is angle E.
M1 — The angle must sit between the two known sides.
A1 — The required angle is angle E.
Question 4 Medium2 marks
A second triangle has all three sides known. Name the condition.
Show answer
Answer: SSS
Three equal sides define congruence.
Criterion is SSS.
M1 — Three equal sides define congruence.
A1 — Criterion is SSS.
Question 5 Hard2 marks
A triangle has two angles and a side that is not between them. Name the condition.
Show answer
Answer: AAS
Side is opposite one of the angles, not between them.
Criterion is AAS.
M1 — Side is opposite one of the angles, not between them.
A1 — Criterion is AAS.
Question 6 Hard2 marks
Two triangles share angles 40, 60, 80. Are they necessarily congruent?
Show answer
Answer: no
Equal angles only prove similarity, not congruence.
Answer is no.
M1 — Equal angles only prove similarity, not congruence.
A1 — Answer is no.
Question 7 Exam-style4 marks
ABCD is a kite with AB = AD and CB = CD. Prove that triangle ABC is congruent to triangle ADC. Give the condition you use.
Show answer
Answer: SSS
AB = AD (given).
CB = CD (given).
AC is a common side of both triangles.
The three pairs of sides are equal, so triangle ABC is congruent to triangle ADC (SSS).
B1 — AB = AD
B1 — CB = CD
B1 — AC is common to both triangles
B1 — Congruent by SSS
Common mistakes
Most lost marks on this topic are one slip.
Calling a non-included side ASA
When the side is not between the two angles, the criterion is AAS
Assuming equal angles mean congruent triangles
Equal angles (AAA) prove similarity, but sides could be different lengths
Using SSA as a valid condition
Two sides and a non-included angle is ambiguous and does not guarantee congruence
Spot the slip
Two angles and a side that is not between them is which condition?
AAS
ASA
SAS
Show the slip
AAS The known side is not between the two angles, so the condition is AAS.
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 three numbered lines, each with a reason in brackets, then the condition on a fourth line — markers look for exactly this.
Mark equal sides with dashes and equal angles with arcs on the diagram as you find them.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.