Theorem 6.5: Two line segments are congruent if and only if they have the same length.

Proof: If the segments are congruent, then it is possible to move one onto the other by an isometry. Since isometries preserve distances, the distance between the endpoints, which is the same thing as the lengths, are the same in both segments.

For the converse, suppose that the two line segments have the same length. Let the first line segment have endpoints   A   and   B,   and let the second have endpoints   A'   and   B'.   Translate   A'   to   A.   Rotate the plane about point   A   through an angle same size as the angle defined by these two segments with their common point at   A.   This will move   B'   onto the ray from   A   through   B.   Since   B'   will now be on the ray from   A   through   B   at the same distance from   A   as   B,   B'   will coincide with   B   by Theorems 2.1 and 2.2. We conclude that the line segment between   A'   and   B'   will have been moved to the line segment between   A   and   B   by Theorem 4.2.

next theorem (6.6)