Axiom:Birkhoff's Axioms

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

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

for all points $A$ and $B$.

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

Postulate III: Postulate of Angle Measure
A set of rays $\left\{{l, m, n, \ldots}\right\}$ through 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 $l$ and $m$ respectively,
 * then the difference $a_m - a_l \pmod {2 \pi}$ of the numbers associated with the lines $l$ and $m$ is $\angle AOB$.

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

then:
 * $d \left({B', C'}\right) = kd \left({B, C}\right)$, $\angle C'B'A' = \pm \angle CBA$, and $\angle A'C'B' = \pm \angle ACB$