Congruent triangles: SSS, SAS, ASA and RHS explained
Learn the congruence conditions for triangles, why AAA does not prove congruence and how to use them in simple proofs, with worked examples.
GCSEHigh SchoolCurriculum fit: GCSE Maths Geometry and measures (Foundation and Higher). US: Geometry (Congruence, including triangle congruence criteria).
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What is it?
Two triangles are congruent if they are exactly the same shape and size. You do not need to check all six measurements (three sides and three angles). A few measurements are enough to prove congruence.
The standard conditions are known by their letters: S for side, A for angle.
Key formulas and ideas
- SSS: all three sides are equal.
- SAS: two sides and the included angle (the angle between them) are equal.
- ASA (or AAS): two angles and a corresponding side are equal.
- RHS (called HL in the US): a right angle, the hypotenuse and one other side are equal.
- AAA (three angles) does not prove congruence: it only proves similarity.
Worked examples
Example 1: SSS
Question: Triangle ABC has AB = 5, BC = 7, AC = 9. Triangle DEF has DE = 5, EF = 7, DF = 9. Are they congruent?
- Compare the sides: AB = DE, BC = EF, AC = DF.
- All three pairs are equal.
Answer: Yes, congruent by SSS
Example 2: SAS
Question: In triangles PQR and XYZ, PQ = XY = 6, QR = YZ = 8 and angle Q = angle Y = 50°. Are they congruent?
- Two sides are equal: PQ = XY and QR = YZ.
- The angle between those sides is equal: angle Q = angle Y.
- This matches SAS.
Answer: Yes, congruent by SAS
Example 3: Why AAA is not enough
Question: Why are two triangles with angles 60°, 60°, 60° not necessarily congruent?
- Both are equilateral, so the angles are equal.
- But one could have sides of 2 cm and the other 5 cm.
- They are similar but not congruent.
Answer: AAA gives similar triangles, not congruent ones
Common mistakes
- Using SSA, which does not prove congruence (except in the right-angled RHS case).
- Writing the angle that is not between the two sides when claiming SAS.
- Listing vertices in a different order for the two triangles.
- Using AAA as a congruence test.
Practice questions
Frequently asked questions
What does 'included angle' mean?
The angle that lies between the two sides you are using.
Why do we prove triangles congruent?
Once triangles are congruent, all their other matching sides and angles are also equal, which lets you prove further facts.
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Reviewed by StudyMorrow editorial review on 5 Nov 2026.