Axiom:Birkhoff's 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 $$\angle A'C'B' = \pm \angle ACB$$