Theorem

Congruence criteria for triangles

Geometry • Triangle • Congruence

Statement

Two triangles are congruent if they agree in SSS (three sides), SAS (two sides and the angle included between them), ASA or AAS (two angles and a corresponding side), or RHS (right angle, hypotenuse and one other side); AAA is not a congruence criterion, since it establishes similarity only.

How to use it

1List every equality you actually have in the two triangles, then read the pattern off that list rather than guessing the rule name.
2A shared side or a shared angle counts as one of the equalities.
3The position of the equal side decides between SAS and ASA: in SAS it lies between the two equal sides' angle, in ASA it lies between the two equal angles.
4Write the correspondence in matching order, $△ A B D ≅ △ C B D$ , so that corresponding parts can be read straight off afterwards.

Worked example

In triangles

A B D

and

C B D

A B = C B

A D = C D

, and

B D

is common. Which criterion applies, and what follows about

∠ A B D

Step-by-step

1
$A B = C B, A D = C D, B D = B D$
Why: The shared side supplies the third pair.
2
$△ A B D ≅ △ C B D$ by SSS
Why: Three pairs of equal sides.
3
$∠ A B D = ∠ C B D$
Why: Corresponding parts of congruent triangles are equal.

Answer

SSS, and

∠ A B D = ∠ C B D

Watch out

SSA is not a criterion — it is the ambiguous case — and neither is AAA. In SAS the equal angle must be the one INCLUDED between the two equal sides; an angle elsewhere proves nothing.

Also known as

SSS, SAS, ASA, AAS, RHS, congruent triangles, congruence rule, CPCT, corresponding parts of congruent triangles, which congruence criterion

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)

2018 (I)

2017 (II)

2017 (I)