Parallel axiom: Two lines must intersect if they meet a transversal so that the sum of two interior angles, on the same side of the transversal, is less than 180 degrees.

Parallel Axiom

Let A, B, C, D be points such that A ≠ B, C ≠ D, the points C and D are not on the line through A and B, and the points B and D are on the same side of the line AC. If the sum of the absolute angle measures a(B,A,C) + a(D,C,A) is less than 180 degrees, then the ray from A through B and the ray from C through D have a point in common.

 

Definition: Two lines L and M are parallel if either (a) L = M or (b) the intersection of L and M is empty (there are no points P such that P lies on both L and M).

Definition: If L and M are two distinct lines, a transversal of L and M is a line T which meets L and M at two distinct points.

Thus the parallel axiom says that if two lines L and M meet a transversal so that the sum of the interior angles on the same side of the transversal is less than 180 degrees, then L and M intersect on that side of the transversal.

C. McCrory 9/17/04