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

then $\ell_1 \parallel \ell_3$.

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

Proof

By contradiction. Suppose $\ell_1 \not\parallel \ell_3$. By the definition of parallelism, this means $\ell_1 \neq \ell_3$ and the two must intersect; let $P$ be the intersection point.

Consider the lines through $P$ parallel to $\ell_2$. On the one hand, since $\ell_1 \parallel \ell_2$ and $P\in\ell_1$, $\ell_1$ is such a line; on the other hand, $\ell_2 \parallel \ell_3$ (parallelism is symmetric) and $P\in\ell_3$, so $\ell_3$ is also such a line.

Hence at least two distinct lines through $P$, namely $\ell_1$ and $\ell_3$, are both parallel to $\ell_2$ — contradicting uniqueness of parallels. So the original assumption fails, and $\ell_1 \parallel \ell_3$. $\blacksquare$

Immediate consequences

• Parallelism is an equivalence relation: reflexivity follows from the definition (every line coincides with itself, hence is parallel to itself); symmetry is a direct consequence of the definition; transitivity is precisely this theorem. Classifying all lines in the plane by "parallel to" partitions them into classes, each called a direction, and "direction" becomes a well-defined notion.

• Direction is preserved under translation: translating the entire figure by some vector sends each line to a line parallel to the original (this can be obtained by the later "draw a parallel through a point"). Combined with this theorem, two parallel lines remain parallel under translation — the foundation for the rigid relations of "parallelograms" later on.

Remarks

The proof uses the symmetry of parallelism ($\ell_2 \parallel \ell_3 \Rightarrow \ell_3 \parallel \ell_2$), which follows directly from the symmetric definition "coincident or non-intersecting" — no separate theorem required.

Historically, the transitivity of parallelism was often listed as a direct corollary of the "parallel postulate"; in our system, the substantive work is handed to uniqueness of parallels, and this theorem only uses contradiction to translate "two parallels" into "two parallels through $P$", at which point uniqueness suffices.