Let $P$ and $Q$ two planes in the Euclidean $3$-dimensional space that intersect in a line $L$.
In the presence of a Cartesian coordinate system, the angle between $P$ and $Q$ can be defined as the angle between any two non-zero vectors $v$ and $w$ that are orthogonal to $P$ and to $Q$, respectively, and such that the angle between $v$ and $w$ is not obtuse. It can be shown with the tools of Analytic Geometry that this angle is the same regardless of the vectors $v$ and $w$ chosen from the many pairs of vectors that satisfy the requirements.
How can the angle between planes be defined in the absence of a Cartesian coordinate system, and without recourse to the tools of Analytic Geometry, but only to Hilbert's axioms? How can this definition be shown to be well-defined?
I expect the definition to go along the following lines.
Choose any point $A$ on $L$, and let $S$ and $T$ denote the lines lying in $P$ and $Q$, respectively, that are perpendicular to $L$ and pass through $A$. Then the non-obtuse angle between $S$ and $T$ is defined to be the angle between $P$ and $Q$.
The real question that I'm interested in is: how can it be shown that this definition is well-defined and is independent of the choice of the point $A$?