#### The SSS congruence test

Suppose first that we are asked to construct a triangle ABC in which

AB = 12 cm and AC = 5 cm.

There are infinitely many ways to do this, because the two sides can remain joined at A, but flap around. Three such triangles are shown below, and they are clearly not congruent.

This shows that just knowing that two pairs of sides are equal is not enough information to establish congruence.

#### Constructing a triangle with three given sides

When all three sides of a triangle are given, however, there is no longer any freedom of movement, and only one such triangle can be constructed up to congruence.

For example, a triangle with side lengths 3 cm, 4 cm and 5 cm is unique up to congruence.