SSS Theorem — Definition, Formula & Examples

The SSS (Side-Side-Side) Theorem states that if all three sides of one triangle are equal in length to the corresponding three sides of another triangle, then the two triangles are congruent.

If $\triangle ABC$ and $\triangle DEF$ satisfy $AB = DE$ , $BC = EF$ , and $AC = DF$ , then $\triangle ABC \cong \triangle DEF$ . That is, a triangle is uniquely determined (up to reflection) by its three side lengths, guaranteeing congruence of all corresponding angles as well.

How It Works

To use SSS, you need to show that each side of one triangle matches exactly one side of the other triangle. No angle measurements are required. Once you establish all three pairs of equal sides, you can conclude the triangles are congruent, meaning every corresponding angle is also equal. Order matters: make sure you match corresponding vertices correctly so that the sides you compare actually occupy the same position in each triangle.

Worked Example

Problem: Triangle PQR has sides PQ = 5 cm, QR = 7 cm, and PR = 9 cm. Triangle XYZ has sides XY = 5 cm, YZ = 7 cm, and XZ = 9 cm. Are the triangles congruent?

Step 1: Compare the first pair of corresponding sides.

PQ = XY = 5 \text{ cm}

Step 2: Compare the second pair of corresponding sides.

QR = YZ = 7 \text{ cm}

Step 3: Compare the third pair of corresponding sides.

PR = XZ = 9 \text{ cm}

Answer: All three pairs of sides are equal, so by the SSS Theorem,

\triangle PQR \cong \triangle XYZ

Why It Matters

SSS is one of the first congruence criteria you encounter in a geometry proof course. Engineers and architects rely on the rigidity of triangles — the fact that three fixed side lengths lock in a unique shape — when designing stable structures like bridges and trusses.

Common Mistakes

Mistake: Matching sides that are not corresponding (e.g., comparing the longest side of one triangle to the shortest side of another).

Correction: Always identify which vertices correspond to each other first, then compare sides that connect matching pairs of vertices.