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, by the intermediate value theorem, 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, angle BAC has a radians, and so does angle 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