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.

next theorem (5.9)