Through a point off a line, exactly one parallel exists (Playfair)

Dependencies: Corresponding/alternate angles ⇔ lines parallel (Corresponding/alternate angles ⇔ lines parallel), dropping a perpendicular from a point to a line (Perpendicular from a point to a line exists and is unique), the triangle exterior-angle inequality (Exterior angle > either non-adjacent interior angle).

Statement

Let $\ell$ be a line, and let $P$ be a point off $\ell$ ($P\notin\ell$). Then there exists exactly one line through $P$ parallel to $\ell$.

Denote this unique parallel by $n$. The statement has two halves: existence — at least one can be constructed; uniqueness — there is no second one. This is the celebrated Playfair's axiom of elementary geometry — equivalent in Euclidean geometry to Euclid's fifth postulate; here we treat it as a theorem to be proved, using only the three already-established results above.