- Points
- Lines
- Planes
- Lie on, contains
- Between
- Congruent

*Axioms of Incidence***Postulate I.1.**- For every two points
`A`,`B`there exists a line`a`that contains each of the points`A`,`B`. **Postulate I.2.**- For every two points
`A`,`B`there exists no more than one line that contains each of the points`A`,`B`. **Postulate I.3.**- There exists at least two points on a line. There exist at least three points that do not lie on a line.
**Postulate I.4.**- For any three points
`A`,`B`,`C`that do not lie on the same line there exists a plane α that contains each of the points`A`,`B`,`C`. For every plane there exists a point which it contains. **Postulate I.5.**- For any three points
`A`,`B`,`C`that do not lie on one and the same line there exists no more than one plane that contains each of the three points`A`,`B`,`C`. **Postulate I.6.**- If two points
`A`,`B`of a line`a`lie in a plane α then every point of`a`lies in the plane α. **Postulate I.7.**- If two planes α, β have a point
`A`in common then they have at least one more point`B`in common. **Postulate I.8.**- There exist at least four points which do not lie in a plane.

*Axioms of Order***Postulate II.1.**- If a point
`B`lies between a point`A`and a point`C`then the points`A`,`B`,`C`are three distinct points of a line, and`B`then also lies between`C`and`A`. **Postulate II.2.**- For two points
`A`and`C`, there always exists at least one point`B`on the line`AC`such that`C`lies between`A`and`B`. **Postulate II.3.**- Of any three points on a line there exists no more than one that lies between the other two.
**Postulate II.4.**- Let
`A`,`B`,`C`be three points that do not lie on a line and let`a`be a line in the plane`ABC`which does not meet any of the points`A`,`B`,`C`. If the line`a`passes through a point of the segment`AB`, it also passes through a point of the segment`AC`, or through a point of the segment`BC`.

*Axioms of Congruence***Postulate III.1.**- If
`A`,`B`are two points on a line`a`, and`A'`is a point on the same or on another line`a'`then it is always possible to find a point`B'`on a given side of the line`a'`such that`AB`and`A'B'`are congruent. **Postulate III.2.**- If a segment
`A'B'`and a segment`A"B"`are congruent to the same segment`AB`, then segments`A'B'`and`A"B"`are congruent to each other. **Postulate III.3.**- On a line
`a`, let`AB`and`BC`be two segments which, except for`B`, have no points in common. Furthermore, on the same or another line`a'`, let`A'B'`and`B'C'`be two segments which, except for`B'`, have no points in common. In that case if and`AB`≈`A'B'` , then`BC`≈`B'C'` .`AC`≈`A'C'` **Postulate III.4.**- If ∠
`ABC`is an angle and if`B'C'`is a ray, then there is exactly one ray`B'A'`on each "side" of line`B'C'`such that∠ . Furthermore, every angle is congruent to itself.`A'B'C'`≅∠`ABC` **Postulate III.5. (***SAS*)- If for two triangles
`ABC`and`A'B'C'`the congruences ,`AB`≈`A'B'` and`AC`≈`A'C'`∠ are valid, then the congruence`BAC`≈ ∠`B'A'C'`∠ is also satisfied.`ABC`≈ ∠`A'B'C'`

*Axiom of Parallels***Postulate IV.1.**- Let
`a`be any line and`A`a point not on it. Then there is at most one line in the plane that contains`a`and`A`that passes through`A`and does not intersect`a`.

*Axioms of Continuity***Postulate V.1. (***Archimedes Axiom*)- If
`AB`and`CD`are any segments, then there exists a number`n`such that`n`copies of`CD`constructed contiguously from`A`along the ray`AB`willl pass beyond the point`B`. **Postulate V.2. (***Line Completeness*)- An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence (Axioms I-III and V-1) is impossible.

*Foundations of Geometry*, D. Hilbert (trans. L. Unger), Open Court Publ.*Roads to Geometry*, E. Wallace & S. West, Prentice-Hall (Sect 2.4 & App. B)

R. I. Campbell, campbell@math.umbc.edu 10 Feb, 2002