Axiom:Birkhoff's Axioms

Axioms
These postulates of Euclidean geometry are all based on basic geometry that can be confirmed experimentally with a ruler and protractor.

Postulate I: Postulate of Line Measure
The set of points $\set {A, B, \ldots}$ on any line can be put into a 1:1 correspondence with the real numbers $\set {a, b, \ldots}$ so that:
 * $\size {b - a} = \map d {A, B}$

for all points $A$ and $B$.

Postulate II: Point-Line Postulate
There is one and only one straight line $l$ that contains any two given distinct points $P$ and $Q$.

Postulate III: Postulate of Angle Measure
The set of half-lines $\set {\ell, m, n, \ldots}$ starting at any point $O$ can be put into 1:1 correspondence with the real numbers $a \pmod {2 \pi}$ so that:
 * if $A$ and $B$ are points (not equal to $O$) of $\ell$ and $m$ respectively,
 * then the difference $a_m - a_\ell \pmod {2 \pi}$ of the numbers associated with the lines $\ell$ and $m$ is $\angle AOB$.

Furthermore, let a point $C$ vary on the half-line $r$ starting at $A$ and passing through $B$.

Let $n$ be the half-line starting at $O$ and passing through $C$, varying as $C$ does.

If $r$ and $\ell$ do not lie on a straight line, then as $\map d {C, A} \to 0$, $a_n \to a_\ell$.

Postulate IV: Postulate of Similarity
Given two triangles $ABC$ and $A'B'C'$ and some constant $k > 0$ such that:
 * $\map d {A', B'} = k \map d {A, B}$, $\map d {A', C'} = k \map d {A, C}$ and $\angle B'A'C' = \pm \angle BAC$

then:
 * $\map d {B', C'} = k \map d {B, C}$, $\angle C'B'A' = \pm \angle CBA$, and $\angle A'C'B' = \pm \angle ACB$