**Theorem 6.6**: Two
angles are
congruent if and only if they
have the same size.

**Proof**: If the
angles are
congruent, then it is possible to
move one onto the other by an invertible
isometry. Since such
isometries preserve the
size of
angles by
Theorem 5.6, the
sizes of the
angles will both be the
same.

For the converse, suppose that* * __/ __*ABC *has the same size as* *__/ __*A*'*B*'*C*'. * *Translate * **A*'* *to* **A*. * *Rotate the
plane about* **A *until* **B*'* *is on
the ray from* **A *through* **B*.* *By Theorem 5.13, there are two
angles having the
ray from* **A *through* **B *as
one arm, and they are both reflections of each other
about the line determined by* **AB*.* *So if* **B*'* *is not on the ray from* **A *through* **B*,* *then
it will be after reflecting
about the line determined by* **AB*.* *In either case it will be possible to move* *__/ __*A*'*B*'*C*'* *to* *__/ __*ABC *by a translation,
rotation, and perhaps a
reflection, so the
angles will be
congruent.