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

Congruent triangles are triangles with the same size and shape: if one can be slid, rotated, or reflected so it lands exactly on the other, the triangles are congruent. You prove congruence in most geometry problems with one of five tests, and you also need to know which common shortcuts fail.

The order of the vertices matters, because it tells you which parts correspond. If $\triangle ABC \cong \triangle DEF$, then matching parts are equal:

and

The statement matches $A$ with $D$, $B$ with $E$, and $C$ with $F$. Writing the letters in the wrong order can match the wrong sides and angles.

When To Use Each Congruence Test

Pick the test that matches the parts you are given. Five tests 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, the one 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 is too, and the side fixes the size.
• AAS (Angle-Angle-Side): if two angles and a non-included side match, the triangles are congruent, for the same reason as ASA.
• HL (Hypotenuse-Leg): applies only to right triangles. If two right triangles share the same hypotenuse and one matching leg, they are congruent. Without the right-angle condition, HL does not apply.

Two famous shortcuts do not prove congruence. AAA only proves the triangles are similar: two equilateral triangles can both have angles $60^\circ$, $60^\circ$, $60^\circ$ but different side lengths. SSA is the ambiguous case: knowing two sides and a non-included angle can produce more than one triangle, or none. The common exception is HL, which works only because the triangles are right triangles.

The Steps For A Proof

1. Mark matching parts

Identify which sides or angles in one triangle correspond to which parts in the other.

2. Check a valid test

Use one of SSS, SAS, ASA, AAS, or HL for right triangles.

3. State the condition clearly

Make sure any extra requirement is met, such as the included angle in SAS or the right-angle condition in HL.

4. Conclude carefully

Once a valid test is satisfied, the triangles are congruent, so all remaining corresponding parts match too.

A Full Worked SAS Example

Suppose you know

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

That is enough for SAS:

So the triangles are congruent:

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.

Where The Steps Trip People Up

The condition-checking step is where most errors happen:

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.

A quick self-check for the HL 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.

Why It Matters

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.