PRINCIPIA · THEOREM

Transitivity of parallelism

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

Statement

Let $\ell_1$ , $\ell_2$ , $\ell_3$ be three lines in the plane. If

\ell_1 \parallel \ell_2 \quad\text{and}\quad \ell_2 \parallel \ell_3,

then $\ell_1 \parallel \ell_3$ .

Here we adopt the convention "coincident or non-intersecting = parallel" — every line is parallel to itself.

$Transitivity of parallelism: \ell_1 \parallel \ell_2 and \ell_2 \parallel \ell_3 ⇒ \ell_1 \parallel \ell_3$

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