Geometry Axioms

A list of geometry axioms starting with the basics of incidence, betweeness, and congruence.

Axiom (Incidence Axioms) The following axioms are called the Incidence Axioms.

    (i) (Line Uniqueness) Given two distinct points and there exists a unique line incident with and

    (ii) (Points On Line) For every line there exist at least two distinct points incident with

    (iii) (Non-Collinear Points) There exists three non-collinear points.

Axiom (Betweenness Axioms) The following axioms are called the Betweenness Axioms.

    (i) (Linearity) If , then and are three collinear points and

    (ii) (Extension) Given any two distinct points and , there exist a point lying on such that

    (iii) (Order) If and are three distinct points on the same line, then one and only one of the points is between the other two.

    (iv) (Separation) Given any line and any three points and not lying on . If and are on the same side of and and are on the same side of then and are on the same side of If and are on opposite sides of and and are on opposite sides of   then and are on the same side of

Axiom (Congruence Axioms) The following axioms are called the Congrunce Axioms.

    (i) (Segment Shift) If and are distinct points and if is any point, then for each ray emanating from there is a unique point on such that and

    (ii) (Segment Congruence) If and then Moreover, every segment is congruent to itself.

    (iii) (Additive) If , and then

    (iv) (Angle Shift) Given and there is a unique ray on a given side of such that

    (v) (Angle Congruence) If and then Moreover, every angle is congruent to itself.

    (vi) (Side Angle Side) If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.

Cite this as:
Geometry Axioms
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/geometry-axioms.html