Parallelism is Transitive Relation

Theorem

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

Stated equivalently, parallelism is a transitive relation.

Proof

 * Three Parallel Lines.png

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 $EF$.

By Parallelism implies Equal Alternate Interior Angles:
 * $\angle AGK = \angle GHF$

By Playfair's Axiom, there is only one line that passes through $H$ that is parallel to $CD$ (namely $EF$).

Therefore the transversal $GK$ cannot be parallel to $CD$ and the two lines must therefore intersect.

The straight line $GK$ also cuts the parallel lines $EF$ and $CD$.

So from Parallelism implies Equal Corresponding Angles:
 * $\angle GHF = \angle GKD$.

Thus $\angle AGK = \angle GKD$.

So from Equal Alternate Interior Angles implies Parallel Lines:
 * $AB \parallel CD$