**Theorem
6.7**: (SSS) Two
triangles are
congruent if and only if their
corresponding sides all
have the same lengths.

**Proof**: If they are
congruent, then it is possible to
move one of the
triangles until it
coincides with the other by an invertible
isometry. Since, by
definition, an isometry
preserves distances, the
sides that will
coincide after the movement will have to all have the same
lengths by Theorem
6.5.

For the converse, assume that for the three
sides in* *triangle* ABC *and* *triangle *A*'*B*'*C*',

Translate * **A*'* *to* **A*,* *and rotate the
plane about* **A *until* **B*'* *coincides with* **B*,* *as in the proof of Theorem 6.5. Then* **C*'* *will have been moved to a point whose
distance from* **A *is* *|*AC*|* *and whose distance from* **B *will be * *|*BC*|.* *That is, it will be on the circle
centered at* **A *of radius * *|*AC*|* *and on the circle
centered at* B *of radius * *|*BC*|.

By the triangle inequality, Theorem 3.5, and Theorem 3.13, there
are two such points, and by
Theorem 4.12, they are
reflections of each other
about the line determined by* **AB*.* *So either* **C*'* *coincides with* **C *or it will after a reflection, and the
triangles are
congruent.