An axiom system for Euclidean plane geometry

We will use a version of the 1923 axiom system of the American mathematician George Birkhoff. (See my history of axioms for plane geometry.)

The basic objects are points, and the basic functions are distance, angle (also called angle measure), and area. Objects such as lines, circles, rays, or triangles are defined in terms of the basic data.

Axioms

Here's a summary of the axioms. For details, see the link for each axiom. The first four axioms are foundational, and they are seldom used after the basic theorems have been proved. On the other hand, the last three axioms are quite useful.

(1) Distance axiom: Distance is symmetric, positive definite, and satisfies the triangle inequality.

(2) Line axiom: Two points determine a line. There exist three points which do not lie in a line.

(3) 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.

(4) Betweenness axiom: Betweenness for points on a line corresponds to betweenness for rays from a point.

(5) Congruence axiom: The side-angle-side congruence property holds for triangles.

(6) 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.

(7) Area axiom: Congruent polygons have the same area. The area of the sum of two polygons is the sum of their areas. The area of a rectangle is its base times its height.
 

C. McCrory 9/17/04