**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.