Modern axiomatisation of Euclid's geometry is due to David Hilbert in his book Grundlagen der Geometrie (The Foundations of Geometry, 1899).
A modern treatment is in Robin Hartshorne, Geometry Euclid and Beyond (2000).
According to Hilbert [see page 4 of the English translation] :
Definition. We will call the system of two points $A$ and $B$, lying upon a straight
line, a segment and denote it by $AB$ or $BA$.
Hilbert states [page 6] :
Definition. A system of segments $AB, BC, CD, \ldots, KL$ is called a broken line joining $A$ with $L$ and is designated, briefly, as the broken line $ABCDE \ldots KL$. The points lying within the segments $AB, BC, CD, \ldots, KL$, as also the points $A, B, C, D, \ldots, K, L$, are called the points of the broken line. In particular, if the point $A$ coincides with $L$, the broken line is called a polygon and is designated as the polygon $ABCD \ldots K$.
The segments $AB, BC, CD, \ldots, KA$ are called the sides of the polygon and the points
$A, B, C, D, \ldots, K$ the vertices. Polygons having $3, 4, 5, \ldots, n$ vertices are called, respectively, triangles, quadrangles, pentagons, ..., $n$-gons.
Thus, triangles are identified by three non-collinear point $A, B, C$.
See also [page 6] :
All of the points of [a line] $a$ which lie upon the same side of [a point] $O$, when taken together, are called the half-ray emanating from $O$. Hence, each point of a straight line divides it into two half-rays.
Thus, considering the segment $AB$ on line $a$, we have that $A$ divides $a$ into two half-rays, one of which contains point $B$, and so all the segment $AB$.
Finally, we have [page 8] :
Definitions. Let be any arbitrary plane and $h, k$ any two distinct half-rays lying
in and emanating from the point $O$ so as to form a part of two different straight lines.
We call the system formed by these two half-rays $h, k$ an angle [...].
Thus, the half-rays containing segments $AB$ and $AC$ form an angle; the same for the couples $BA$ with $BC$ and $CB$ with $CA$.
