Straight Angle Theorem: If the points A, B, C are distinct, then B is between A and C if and only if a(ABC) = 180.

Proof:

(1) First we assume that B is between A and C, and prove that @ABC = 180 degrees.

The point B is between A and C means that B lies on the line segment AC. Since A, B, and C are distinct points, B is not an endpoint of the segment AC. Consider the degenerate triangle ABC. Since B lies on Ray AC, the angle CAB is 0 (angle axiom). By the Side-Angle-Side congruence axiom, the triangle ABC is negatively congruent to itself. (AB = AB, AC = AC, @CAB = - @CAB.) Therefore @ABC = -@ABC. Since angle measure is mod 360 degrees, this implies that @ABC = 180 degrees or @ABC = 0 degrees. If @ABC = 0, then C would lie on Ray BA (by the angle axiom). This contradicts that B is between A and C. Thus @ABC = 180 degrees.

(2) Next we assume that @ABC = 180 degrees, and prove that B is between A and C.

Let L be a line, and let B' be a point on L. Let A' and C' be points on L such that A'B' = AB, B'C' = BC, and B' is between A' and C'. (The points A' and C' exist by the distance and line axioms.) By the first part of the proof, we have that @A'B'C' = 180. Thus by the Side-Angle-Side congruence axiom, triangle ABC is congruent to triangle A'B'C'. (AB = A'B', BC = B'C', @ABC = @A'B'C'.) Therefore @CAB = @C'A'B'. But @C'A'B' = 0 since B' lies on Ray A'C' (angle axiom). So @CAB = 0. Therefore B lies on Ray AC (angle axiom). Similarly, @BCA = 0, so B lies on Ray CA. Therefore B lies on the segment AC, i.e. B is between A and C.