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 $$AB \parallel CD$$.