Congruence and Congruency Criteria — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers when two shapes are identical, and how to prove it. By the end of this guide you should be able to say what congruence means, recognise congruent shapes, and state the four criteria for congruent triangles.

You should also be able to set out a formal congruence proof with reasons, identify corresponding vertices, explain why one apparent criterion does not work, and use congruence to prove further results about a figure.

The organising idea is that proving congruence needs three facts, and at least one of them must be a side. Angles alone fix the shape but not the size — two triangles with identical angles can differ wildly in scale, which makes them similar rather than congruent. That is exactly why every valid criterion contains an S, and why the tempting "AAA" is not on the list. Remembering that one requirement tells you what to hunt for in a proof: find your three facts, and make sure a side is among them.

Key terms and definitions

Congruent — identical in shape and size, so all corresponding sides and angles are equal.

Similar — the same shape but possibly a different size.

Corresponding — occupying matching positions in the two shapes.

Included angle — the angle between two named sides.

Included side — the side between two named angles.

Hypotenuse — the side opposite the right angle in a right-angled triangle.

Criterion — one of the accepted sets of three facts sufficient to prove congruence.

Core concepts

What congruence means

Two shapes are congruent if one can be placed exactly on top of the other, allowing for turning and flipping.

So congruent shapes may be rotated or reflected relative to each other and still be congruent. They are not required to be the same way up, which is why questions often present one triangle turned round to make the correspondence harder to see.

Translation, rotation and reflection all produce congruent images; enlargement does not, unless the scale factor is 1.

Congruence is the special case of similarity where the scale factor is exactly 1. Every pair of congruent shapes is therefore similar, but not the reverse.

The four criteria for triangles

Three facts prove two triangles congruent, provided they form one of these four sets.

SSS — three pairs of equal sides.

SAS — two pairs of equal sides and the included angle, meaning the angle between those two sides.

ASA — two pairs of equal angles and a corresponding side. The side is usually the included one, between the two angles, though any corresponding side works once two angles match, since the third angle follows from the 180° total. Some texts write that variant as AAS.

RHS — in right-angled triangles, the right angle, the hypotenuse and one other side.

The word included is doing real work in SAS. Two sides and a non-included angle is not a valid criterion, and it genuinely fails: two different triangles can be built from the same two sides and the same non-included angle.

Why AAA does not work

Three equal angles fix the shape but say nothing about size.

A triangle with angles 30°, 60° and 90° and sides of 1, √3 and 2 has exactly the same angles as one with sides 10, 10√3 and 20. They are similar, not congruent.

This is the reason every criterion contains at least one S, and it is worth stating explicitly in an exam answer when a question asks why angles alone are not enough.

Identifying corresponding vertices

Before writing a proof, work out which vertex of one triangle matches which of the other.

Match by the facts you have: equal sides go with equal sides, and the angle between two sides in one triangle corresponds to the angle between the matching sides in the other.

Once identified, name the triangles in matching order. Writing "triangle ABC is congruent to triangle PQR" asserts that A matches P, B matches Q and C matches R, so the order carries information and must be right.

This also means that after proving congruence, you may conclude that any other pair of corresponding parts is equal — which is usually the real purpose of the proof.

Setting out a proof

A congruence proof is a short, formal list. Three statements of equality, each with a reason, followed by the criterion and the conclusion.

For example:

AB = PQ (given), angle BAC = angle QPR (given), AC = PR (given). Therefore triangle ABC is congruent to triangle PQR by SAS, since the equal angle lies between the two equal sides.

Naming the criterion is essential and carries its own mark. A proof that establishes three correct facts but never says "SAS" is incomplete.

Reasons you can use

The three facts rarely come free; usually at least one must be justified from the figure.

Common reasons include: given or marked on the diagram; common side, where the two triangles share an edge; vertically opposite angles are equal; alternate or corresponding angles, where parallel lines are marked; base angles of an isosceles triangle are equal; radii of the same circle are equal; and properties of a named shape, such as opposite sides of a parallelogram being equal.

The common side is the one most often overlooked. Where two triangles overlap along an edge, that edge is equal to itself and counts as one of your three facts.

Using congruence to prove something else

Most exam questions do not stop at congruence; they use it to establish a further result.

The structure is always the same: prove the triangles congruent, then state that the parts you want are corresponding parts of congruent triangles and therefore equal.

For instance, to prove the diagonals of a parallelogram bisect each other, show that two of the triangles formed are congruent by ASA, using alternate angles and the equal opposite sides. It then follows that the corresponding segments of the diagonals are equal, which is what bisecting means.

Writing that final sentence matters. The congruence alone does not answer the question; the deduction from it does.

Congruence in quadrilaterals and other shapes

The four criteria are stated for triangles, but other shapes are handled by splitting them into triangles with a diagonal.

To show two quadrilaterals are congruent, divide each with a diagonal and prove the resulting triangles congruent in pairs.

For shapes generally, congruence still means all corresponding sides and angles are equal — there is simply no shortcut criterion beyond triangles.

Worked examples

Example 1: Choosing the criterion

Two triangles have sides of 7 cm and 9 cm with a 40° angle between those sides in each. Are they congruent, and by which criterion?

Two pairs of equal sides are given, and the equal angle lies between them.

That is the included angle, so the criterion is SAS.

The triangles are congruent. Had the 40° been at a different vertex — not between the 7 cm and 9 cm sides — the facts would be two sides and a non-included angle, which proves nothing.

Example 2: A proof using a common side

In a kite ABCD, AB = AD and CB = CD. Prove that triangle ABC is congruent to triangle ADC.

AB = AD (given).

CB = CD (given).

AC = AC (common side, shared by both triangles).

Three pairs of equal sides, so triangle ABC is congruent to triangle ADC by SSS.

The common side is the fact that makes the proof work, and it is the one candidates most often fail to spot.

Example 3: Using congruence to prove a further result

In the kite above, use the congruence to show that angle ABC equals angle ADC.

The triangles ABC and ADC have been proved congruent by SSS.

Angle ABC and angle ADC are corresponding angles of those congruent triangles — B corresponds to D, as the naming order shows.

Corresponding parts of congruent triangles are equal, so angle ABC = angle ADC.

That closing sentence is the answer to the question; the congruence was only the means to it.

Common mistakes and how to avoid them

Using AAA. Equal angles give similarity, not congruence. Every valid criterion contains a side.

Ignoring the word "included" in SAS. Two sides and a non-included angle is not a criterion.

Forgetting the common side. Where two triangles share an edge, that edge is equal to itself.

Naming the triangles in the wrong order. ABC congruent to PQR asserts A matches P, and the order is part of the answer.

Giving facts without reasons. Each of the three statements needs a justification.

Omitting the criterion. State SSS, SAS, ASA or RHS explicitly.

Stopping at the congruence. If the question asks for something further, add the sentence deducing it from corresponding parts.

Exam technique for "Congruence"

List your three facts with a reason beside each, then name the criterion, then state the conclusion. That structure maps directly onto the mark scheme.

Look for the common side first whenever two triangles overlap. It is free and it is frequently the missing third fact.

Check the diagram markings — dashes for equal sides, arcs for equal angles, the square for a right angle — and use only what they justify.

Identify corresponding vertices before writing anything, and keep the naming order consistent throughout.

If the question asks you to prove something beyond congruence, finish with the sentence about corresponding parts of congruent triangles.

Where a question asks why a particular set of facts is insufficient, name the criterion it fails to meet and explain that the shapes would be similar rather than congruent.

Quick revision summary

Congruence needs three facts, and at least one must be a side — which is why every criterion contains an S and why AAA proves only similarity.

The four criteria: SSS; SAS with the angle included between the two sides; ASA with a corresponding side; and RHS for right-angled triangles, using the right angle, the hypotenuse and another side.

Congruent shapes may be rotated or reflected, so they need not be the same way up. Congruence is similarity with a scale factor of 1.

Name triangles in matching order: ABC congruent to PQR asserts A matches P.

Useful reasons include given, common side, vertically opposite angles, alternate angles, base angles of an isosceles triangle, and radii of the same circle. The common side is the one most often missed.

A proof is three facts with reasons, then the named criterion, then the conclusion.

To prove something further, add that the parts are corresponding parts of congruent triangles and are therefore equal.