**Theorem 5.8**: The
number of radians in a
right angle is half of the
number of radians in a
straight angle.

**Proof**: In the figure,

__/ __*BAC * is the right
angle, and *C*' is the reflection of *C *about the line containing *A *and *B*. By
the definition of a reflection, __/__*CAC*' is
a straight angle. Since
isometries preserve
line segments by
Theorem 4.2, *AC *goes to *AC*'. Since we are reflecting about *AB*, *AB *remains fixed. Conclude that / *BAC*' is the image of / *BAC*. Since isometries preserve the
size of
angles by
Theorem 5.6, we conclude that the two
angles have the same number of
radians. But since they add up
to a straight angle, and,
by Theorem 5.4, the
arc lengths, hence the number
of radians add in this case, we
have two angles with the same
number of radians adding up to a
straight angle. They will
then both need to have half as many
radians as a
straight angle.