Angle axiom: The angle function is antisymmetric, it depends only on rays, and it is additive. Angles of every measure occur with a given ray as initial side.

Angle measure is a function which assigns, to every ordered triple of points (A,O,B) with A and B different from O, a real number (mod 360), denoted @(A,O,B).

It is useful to think of the real numbers mod 360 as corresponding to points on a circular protractor, with each point on the circle corresponding to a degree measure mod 360. Thus, for example, 30 degrees is the same angle measure as 390 degrees or - 330 degrees.

Angle axiom

1. Antisymmetry: For all A, B, O,  with A and B different from O,

        @(A,O,B) = - @(B,O,A).

2. Ray: For all A, B, O, with A and B different from O,

        @(A,O,B) = 0 if and only if B lies on the ray OA.

3: Addition: For all A, B, C, O,  with A, B, and C different from O,

        @(A,O,B) = @(A,O,C) + @(C,O,B).

4. Completeness: For all A, O, with A different from O, and all real numbers x there exists B different from O such that

        @(A,O,B) = x (mod 360).

Because of property (2), the angle measure @(A,O,B) depends only on the two rays Ray(OA) and Ray(OB). For this reason, many books define an "angle" to be the union of two rays, the "sides" of the angle. For short we will often write @(AOB) instead of @(A,O,B).

In GSP, angle measure is written m∠AOB.) In GSP there are two basic options for how to measure an angle, degrees or directed degrees:

If @(A,O,B) is not 180, let @'(A,O,B) be the number between -180 and 180 such that @'(A,O,B) = @(A,O,B) (mod 360). We call @'(A,O,B) the proper angle measure. This is exactly the same as the GSP directed degrees measure.

Let a(A,O,B) = | @'(A,O,B) |, the absolute value of @'(A,O,B). We call a(A,O,B) the absolute angle measure. If @(A,O,B) = 180, we define a(A,O,B) = 180. This is exactly the same as the GSP degrees measure.

Comment on Birkhoff's Protractor Axiom

I've introduced the completeness property as a simplified version of Birkhoff's "Protractor axiom." It is easy to see that the protractor axiom is implied by my angle axiom (parts 1-4). Birkhoff's axiom is the following:

For every point O there is a bijection p from the set of rays with endpoint O to the set of real numbers (mod 360) such that if A and B are points not equal to O, then

            @(AOB) = p(ray(OB)) - p(ray(OA)).

The function p is called a protractor function.

In his high school geometry book (Basic Geometry, by George David Birkhoff and Ralph Beatley, first published in 1940), Birkhoff states this axiom as follows:

"All half-lines having the same end-point can be numbered so that number differences measure angles."

In almost all high school geometry books -- including Birkhoff's! -- absolute angle measure is used, even though signed angles are needed for trigonometry. The following properties of absolute angle measure are consequences of the angle axioms - but they cannot replace the angle axioms unless additional axioms are added.

Properties of absolute angle measure

1. Symmetry: For all A, B, O,  with A and B different from O,

        a(A,O,B) = a(B,O,A).

2. Ray: For all A, B, O, with A and B different from O,

        a(A,O,B) = 0 if and only if B lies on the ray OA.

3: Addition: For all A, B, C, O,  with A, B, and C different from O, if Ray(OC) is between Ray(OA) and Ray(OB), or if Ray(OA) and Ray(OB) are opposite, then

        a(A,O,B) = a(A,O,C) + a(C,O,B).

Note: Two rays with the same endpoint are opposite if their union is a line. Betweenness for rays is defined using the angle measure @. (See the betweenness axiom.)

4. Completeness: For all A, O, with A different from O, and all real numbers x, with 0 < x < 180 there exist points B and C on opposite sides of the line OA such that

        a(A,O,B) = x

        a(A,O,C) = x

Note: The sides of a line are also defined using the angle measure @. (See the betweenness axiom.)