Vertical angle theorem: Suppose that A, O, A' are collinear and B, O, B' are collinear, with O between A and A' and O between B and B'. Then the angle measures of AOB and A'OB' are equal: @(A,O,B) = @(A'O,B').

Segment addition theorem: The point C is between the points A and B if and only if AB = AC + CB.

This theorem says that equality occurs in the triangle inequality AB ≤ AC + CB (part of the distance axiom) if and only if the three vertices A, B, C of the triangle are collinear (the triangle is "degenerate"), with C on the segment AB.

Existence and uniqueness of perpendicular: For every line L and every point A there exists a unique line through A perpendicular to L.

This theorem adds to the existence of perpendicular theorem (basic theorems, set 1) the statement that there is only one line through A perpendicular to L.

Existence and uniqueness of parallel: Given a line L and a point P not on L, there exists a unique line M through P parallel to L.

This theorem adds to the existence of parallel theorem (basic theorems, set 2) the statement that there is only one line through P parallel to L.

This existence and uniqueness theorem often occurs as the "parallel axiom" in Euclidean geometry books.

Parallel lines and distance: If two lines L and M are parallel then every point on L is the same distance from M. If two distinct points on L are the same distance from M, then L and M are parallel.

Recall that the distance from a point P to a line M is the length of the length of the segment PQ, where Q is on M and the line PQ is perpendicular to M.

Exterior angle theorem: An exterior angle of a triangle equals the sum of the angles at the other two vertices of the triangle. In other words, if ABC is a triangle, and D lies on ray BA, then a(DAC) = a(BCA) + a(ABC).