Parallelism is Transitive Relation

Theorem
Infinite straight lines parallel to the same straight line are parallel to the same straight line are parallel to each other.

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 $AB \parallel CD$.