Playfair's axiom

Remarks

In absolute geometry, at least one such line always exists: drop the perpendicular from $P$ to $\ell$ and take the perpendicular to it at $P$. So, over those axioms, Playfair's axiom says that the parallel through $P$ exists and is unique. It is the parallel axiom in Hilbert's system, and it is equivalent to Euclid's fifth postulate over the other axioms. In the hyperbolic plane it fails: through $P$ there are infinitely many lines that do not meet $\ell$. In the real plane $\R^2$ with $\ell = \set{(x, 0) : x \in \R}$ and $P = (0, 1)$, the only line through $P$ missing $\ell$ is $y = 1$.

Going deeper

For a student

Playfair's axiom is the form of the parallel postulate that one uses in practice. It speaks only of incidence, and not of angles, so it also makes sense in an affine plane, where it is one of the defining axioms.