SSS criterion

The most direct way to confirm two triangles are congruent is to check all three sides. If they match, the rest follows.

Idea

SSS (Side-Side-Side) criterion. If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.

In symbols: if $AB = DE$, $BC = EF$, $CA = FD$, then $\triangle ABC \cong \triangle DEF$.

Why does it work? Once you fix three side lengths, the triangle is rigid , there is essentially only one way to assemble it (up to flipping). The three sides force the three angles, the position of each vertex, everything.

This is why triangles are used in bridges and trusses: they don't deform when their sides are fixed. A rectangle made of bars can become a parallelogram (the angles can shift), but a triangle of bars cannot change shape.

Caution. Make sure the sides match in pairs. The side $5$ cm in one triangle must match the side $5$ cm in the other, not the side $6$. Always identify which side corresponds to which.

Worked examples

Example 1. In triangle $ABC$, $AB = 5, BC = 7, CA = 6$. In triangle $PQR$, $PQ = 5, QR = 7, RP = 6$. Are they congruent?

By SSS, yes. $\triangle ABC \cong \triangle PQR$ with $A \leftrightarrow P$, $B \leftrightarrow Q$, $C \leftrightarrow R$.

Example 2. Two triangles have sides $4, 5, 6$ and $4, 5, 7$. Congruent?

No , the third sides differ.

Example 3. A triangle has sides $3, 4, 5$. So does another. Are they necessarily congruent?

By SSS, yes.

Example 4. Why does the SSS criterion not work for quadrilaterals?

A quadrilateral with sides $4, 4, 4, 4$ could be a square or a rhombus (or many shapes in between). Four sides alone do not fix the angles. (A diagonal also needs to be specified to lock it.)

Try it yourself

1. State the SSS criterion in your own words.
2. Triangles have sides $5,5,5$ and $5,5,5$. Congruent?
3. Triangles have sides $3,4,5$ and $5,4,3$. Congruent?
4. Two triangles share two equal sides but the third differs. Congruent?
5. Two equilateral triangles with sides $7$ each. Congruent?
6. Two scalene triangles with sides $4, 5, 6$. Congruent?
7. Why is the triangle 'rigid' once its three sides are fixed?
8. Sketch two triangles and label sides so that SSS applies.

Activity

Cut out two strips of $5$ cm, two of $6$ cm and two of $7$ cm. Form two triangles independently. You will find both triangles are identical (up to flipping) , SSS in action.