**Theorem 5.13**: (The Protractor
Axiom) Let* **x *be a positive number less than pi. Let* A *and* **B *be two points. Then there are
exactly two lines going
through point* A, *which make an
angle of* **x** * radians with the
ray from* **A *through* **B*.* *In
one, the angle would be measured clockwise from* **A*,* *and in the
other, the angle would be measured counterclockwise from* **A*.* *The two angles are reflections of
each other about the line
determined by* **AB*.

**Proof**: Translate * **A *to the origin, and rotate about
the origin until* B *falls on the negative* **x*-axis. By Theorem 5.9, the
arc length along the
unit circle from* *(-1, 0)* *to a point* *(*s*, *t*)* *is given by

which will be a function of the* **x-*coordinate of the point whether the
point is on the upper
half circle or the lower
half circle. Since* **A*(*s*)* *is
continuous by Theorem 5.10, we can use the
intermediate value theorem to conclude that, if

then there is a real number* **r *between* *-1* *and* *1* *such that

Now there are two points
on the circle which have* **r *as
an* **x-*coordinate, one on the upper half circle and one on the
lower half circle by
Theorem 3.2. Let* **C *be the point on the upper
half circle and let* **D *be the point on the lower
half circle. Then by the
definition of radian measurement,* **m*__/ __*BAC* = *a * radians, and so
is* **m*__/ __*BAD*.* *One would get to point * **C *by going around the circle
clockwise, and to
point * **D *by going around it counterclockwise. By
Theorem 4.10,* **C *and* **D *are reflections of each other
about the line determined by* **AB*.* *By Theorem 4.2, the
line determined by* **AC *is then
the reflection of the
line determined by* **AD*.

6. Parallel Lines and Similar and Congruent Triangles