Congruent Triangles — SSS, SAS, ASA, AAS & HL

Congruent triangles are triangles with the same size and shape. In most geometry problems, you prove congruence with one of five tests: SSS, SAS, ASA, AAS, or HL. You also need to know which common shortcuts fail, especially AAA and SSA.

If one triangle can be slid, rotated, or reflected so that it lands exactly on the other, the triangles are congruent. The order of the vertices matters because it tells you which parts correspond.

What congruent triangles mean

If $\triangle ABC \cong \triangle DEF$, then matching parts are equal:

$$\triangle ABC \cong \triangle DEF$$

This means

$$AB = DE,\ BC = EF,\ AC = DF$$

and

$$\angle A = \angle D,\ \angle B = \angle E,\ \angle C = \angle F$$

The statement $\triangle ABC \cong \triangle DEF$ matches $A$ with $D$, $B$ with $E$, and $C$ with $F$. If you write the letters in the wrong order, you can match the wrong sides and angles.

Triangle congruence tests that work

SSS: Side-Side-Side

If all three side lengths match, the triangles are congruent.

Three side lengths determine one triangle, up to reflection.

SAS: Side-Angle-Side

If two sides and the included angle between them match, the triangles are congruent.

The condition is the included angle. It must be the angle formed by the two known sides.

ASA: Angle-Side-Angle

If two angles and the included side between them match, the triangles are congruent.

Once two angles are fixed, the third angle is fixed too. The side fixes the triangle's size.

AAS: Angle-Angle-Side

If two angles and a non-included side match, the triangles are congruent.

This works for the same reason as ASA: two angles fix the shape, and one side fixes the size.

HL: Hypotenuse-Leg

HL applies only to right triangles. If two right triangles have the same hypotenuse and one matching leg, they are congruent.

Without the right-angle condition, HL does not apply.

A worked SAS example

Suppose you know

$$AB = DE = 7,\quad AC = DF = 5,\quad \angle A = \angle D = 40^\circ$$

and in each triangle the known angle is between the two known sides.

That is enough for SAS:

$$AB = DE$$ $$AC = DF$$ $$\angle A = \angle D$$

So the triangles are congruent:

$$\triangle ABC \cong \triangle DEF$$

After that, every corresponding part matches. For example, $BC = EF$ and $\angle B = \angle E$.

The key observation is positional, not numerical. The equal angle sits between the equal sides, so this is SAS and not SSA.

Why AAA and SSA are not enough

AAA

AAA does not prove congruence. It only proves the triangles are similar.

For example, two equilateral triangles can both have angles $60^\circ$, $60^\circ$, and $60^\circ$ but different side lengths.

SSA

SSA is not a valid congruence test in general. Knowing two sides and a non-included angle can produce more than one triangle, or sometimes no triangle at all.

That is why SSA is called the ambiguous case. The common special exception is HL, which works only because the triangles are right triangles.

Common mistakes with triangle congruence

1. Using SAS when the known angle is not between the two known sides.
2. Using HL on triangles that are not stated or shown to be right triangles.
3. Assuming AAA proves congruence instead of similarity.
4. Matching the wrong vertices when writing a congruence statement.

Where congruent triangles are used

Congruent triangles appear in geometry proofs, construction problems, coordinate geometry, and design work where matching pieces must fit exactly.

They are especially useful when a problem gives only a few lengths or angles but asks for more. Once you prove congruence, you can use the rest of the corresponding parts with confidence.

Try a similar case

Take two right triangles with hypotenuse $10$ and one leg $6$. HL proves they are congruent only if both triangles are right triangles. Remove that condition and the argument fails, which is a good way to remember why HL is a special case.