**Theorem 5.6**: The number of
radians in an
angle is preserved by
invertible
isometries.

**Proof**: Since the number of
radians is figured using
radii and
lengths of
arcs, and since
lengths of
arcs and
radii are preserved by
such isometries
by Theorem 5.3, the number of
radians in an
angle will be
preserved by such
isometries.