Axioms

The School Mathematics Study Group (SMSG), USA, developed an axiomatic system designed for use in high school geometry courses. We use the SMSG axioms for plane Euclidean geometry in the form given in the Mathematics Framework for California Public Schools, p.288.

The system does satisfy all the requirements for Euclidean geometry; that is, all the theorems in Euclidean geometry can be derived from the system. The axioms are not independent of each other. The lack of independence of the axiomatic system allows high school students to more quickly study a broader range of topics.

Undefined Terms: Point, Line and Plane.

Axioms:

  1. Two points A and B determine a unique line, to be denoted by AB.

  2. (The Distance Axiom). To every pair of distinct points there corresponds a unique positive number, called their distance. This distance satisfies the requirement of the next axiom.

  3. (The Ruler Axiom). Every line can be put in one-one correspondence with the real numbers so that if P and Q are two points on the line, then the absolute value of the difference of the corresponding real numbers is the distance between them.

  4. (The Ruler Placement Axiom). Given two points P and Q on a line, the correspondence with real numbers in the preceding axiom can be chosen so that P corresponds to zero and Q corresponds to a positive number.

  5. There are at least three noncollinear points.

  6. (The Plane Separation Axiom). Given a line L. Then the points not on L form two convex sets, and any line segment AB joining a point A in one set and a point B in the other must intersect L. The convex sets are called the half-planes determined by L.

  7. (The Angle Measurement Axiom). To every ABC there corresponds a real number between 0 and 180, to be denoted by mABC, called the measure of the angle.

  8. (The Angle Construction Axiom). Given a line AB and a half-plane H determined by AB, then for every number r between 0 and 180, there is exactly one ray AP in H so that mPAB = r.

  9. (The Angle Addition Axiom). If D is a point in the interior of BAC, then
    mBAC = mBAD + mDAC.

  10. (The Angle Supplement Axiom). If two angles form a linear pair, then their measures add up to 180.

  11. SAS Axiom for congruence of triangles.

  12. (The Parallel Axiom). Through a given external point, there is at most one line parallel to a given line.

  13. (The Area Axiom). To every polygonal region, there corresponds a unique positive number, called its area, with the following properties: (i) congruent triangles have the same area; (ii) area is additive on disjoint unions; and (iii) the area of a rectangle is the product of the lengths of its sides.

  14. SSS Axiom for congruence of triangles.

  15. ASA Axiom for congruence of triangles.

  16. (The AA Axiom for Similarity). Two triangles with two pairs of angles equal are similar.