What you'll learn
Congruence means two shapes are exactly the same size and shape. In this guide you will learn what congruent shapes are, the four conditions for proving triangles congruent (SSS, SAS, ASA, RHS), how to set out a congruence proof, and how congruence is used to prove geometric facts. These are key higher-tier geometry reasoning skills.
Key terms and definitions
Congruent — exactly the same shape and size (one can be placed exactly on the other).
SSS — three sides equal.
SAS — two sides and the included angle equal.
ASA — two angles and the included side equal.
RHS — right angle, hypotenuse and one other side equal.
Included angle/side — the angle between two sides, or the side between two angles.
Core concepts
What congruence means
Two shapes are congruent if they are identical in shape and size — corresponding sides and angles are equal — even if one is rotated or reflected. Congruent shapes can be mapped onto each other exactly by translation, rotation or reflection. This differs from similar shapes, which have the same shape but possibly different sizes.
The four congruence conditions for triangles
To prove two triangles are congruent, you need one of these four sets of equal parts:
- SSS — all three sides equal.
- SAS — two sides and the included angle (the angle between them) equal.
- ASA — two angles and the included side (the side between them) equal. (AAS, two angles and a corresponding side, also works.)
- RHS — in right-angled triangles, the right angle, hypotenuse and one other side equal.
Note that SSA (two sides and a non-included angle) does not guarantee congruence.
Setting out a congruence proof
To prove congruence, state the equal parts clearly with reasons (e.g. "given", "common side", "vertically opposite angles"), then name the condition (SSS, SAS, ASA or RHS) you have satisfied. Use the correct order of vertices so corresponding parts match.
Using congruence to prove facts
Once triangles are proved congruent, corresponding parts are equal, which lets you prove other results — for example, that a shape is a parallelogram, that a line bisects an angle, or that two lengths are equal. This is a common style of higher-tier question.
Worked examples
Example 1: Identifying the condition
Two triangles have all three pairs of sides equal. Are they congruent?
Yes — this satisfies SSS, so the triangles are congruent.
Example 2: SAS
Two triangles share two sides of 5 cm and 7 cm with a 40° angle between them. Congruent?
Yes — two sides and the included angle are equal, satisfying SAS.
Example 3: Why SSA fails
Two triangles have two equal sides and an equal non-included angle. Are they necessarily congruent?
No — SSA is not a valid condition; two different triangles can fit this, so congruence is not guaranteed.
Common mistakes and how to avoid them
Using SSA. Two sides and a non-included angle do not prove congruence.
Wrong "included" part. SAS needs the angle between the two sides; ASA needs the side between the two angles.
Confusing congruent and similar. Congruent = same size; similar = same shape, possibly different size.
Not giving reasons. State why each pair of parts is equal.
Mismatched vertices. Order the vertices so corresponding parts line up.
Exam technique for Congruence
Check for one of SSS, SAS, ASA (AAS) or RHS.
Identify included angles/sides correctly.
State each equal part with a reason, then name the condition.
Avoid SSA as it does not prove congruence.
Use congruence to deduce further equal sides or angles.
Quick revision summary
Two shapes are congruent if they are exactly the same shape and size, with all corresponding sides and angles equal (even if rotated or reflected) — unlike similar shapes, which can differ in size. To prove triangles congruent, satisfy one of four conditions: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side, with AAS also valid), or RHS (right angle, hypotenuse and one other side). Crucially, SSA does not work. Set out a proof by stating each equal part with a reason (given, common, vertically opposite, etc.), then naming the condition met, ordering vertices so corresponding parts match. Once congruent, corresponding parts are equal, which proves further results such as equal lengths or that a shape is a parallelogram. The common errors are using SSA, mistaking the included part, confusing congruent with similar, and omitting reasons. Check for a valid condition, identify included parts, justify each step, and use congruence to deduce more.