Birkhoff's axioms

In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry.

Birkhoff's axiomatic system was utilized in the secondary-school textbook by Birkhoff and Beatley. These axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known as SMSG axioms. A few other textbooks in the foundations of geometry use variants of Birkhoff's axioms.

Postulates
The distance between two points $A$ and $B$ is denoted by $d(A, B)$, and the angle formed by three points $A, B, C$ is denoted by $∠ ABC$.

Postulate I: Postulate of line measure. The set of points ${A, B, ...}$ on any line can be put into a 1:1 correspondence with the real numbers ${a, b, ...}$ so that $|b &minus; a| = d(A, B)$ 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. The set of rays ${ℓ, m, n, ...}$ through any point $O$ can be put into 1:1 correspondence with the real numbers $a (mod 2π)$ so that if $A$ and $B$ are points (not equal to $O$) of $ℓ$ and $m$, respectively, the difference $a_{m} &minus; a_{ℓ} (mod 2π)$ of the numbers associated with the lines $ℓ$ and $m$ is $∠ AOB$. Furthermore, if the point $B$ on $m$ varies continuously in a line $r$ not containing the vertex $O$, the number $a_{m}$ varies continuously also.

Postulate IV: Postulate of similarity. Given two triangles $ABC$ and $A'B'C'$ and some constant $k > 0$ such that $d(A', B' ) = kd(A, B), d(A', C' ) = kd(A, C)$ and $∠ B'A'C' = ±∠ BAC$, then $d(B', C' ) = kd(B, C), ∠ C'B'A' = ±∠ CBA$, and $∠ A'C'B' = ±∠ ACB$.