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.
next theorem (5.8)