Parallelism is Transitive

Theorem

As Euclid defined it:

Straight lines parallel to the same straight line are also parallel to one other.

(The Elements: Book I: Proposition $30$)

Stated equivalently, parallelism is a transitive relation.

Proof

Let the straight lines $AB$ and $CD$ both be parallel to the straight line $EF$.

Let the straight line $GK$ be a transversal that cuts the parallel lines $AB$ and $EF$. It follows that $\angle AGK = \angle GHF$.

By Playfair's Axiom, there is only one line that passes through $H$ that is parallel to $CD$ (namely $EF$), so the transversal $GK$ cannot be parallel to $CD$ and the two lines must therefore intersect.

Since the straight line $GK$ also cuts the parallel lines $EF$ and $CD$, it also follows that $\angle GHF = \angle GKD$.

Thus, $\angle AGK = \angle GKD$, so finally we have $AB \parallel CD$.

$\blacksquare$

Historical Note

This is Proposition 30 of Book I of Euclid's The Elements.

Note that while this result applies to all parallel lines in Euclidean geometry, this proof is only valid when all three lines are in the same plane.