**Theorem 5.7**: The number of
radians in a
straight angle is half of
the number of radians in a full
circle.

**Proof**: Given a
straight angle, draw a
circle centered at the
vertex of the
straight angle. If we
reflect the
plane about the
line which forms the
straight angle, one
half of the
circle moves onto the
other half of the
circle by
Theorem 4.10. Since a
reflection is an
isometry, by
Theorem 4.8, which is
invertible, by Theorem
4.9, and invertible
isometries preserve the
size of
angles, by
Theorem 5.6, we conclude that the two half circles measure
angles of equal
size which add up to the number
of radians in a full
circle, and, hence,
the number of radians in the
straight angle will be
half of the number of radians in
a full circle.