Parallelism is Symmetric Relation

Theorem
Let $S$ be the set of straight lines in the plane.

For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$.

Then $\parallel$ is a symmetric relation on $S$.

Proof
Let $l_1 \parallel l_2$.

By definition of parallel lines, $l_1$ does not meet $l_2$ when produced indefinitely.

Hence $l_2$ similarly does not meet $l_1$ when produced indefinitely.

That is:
 * $l_2 \parallel l_1$

Thus $\parallel$ is seen to be symmetric.

Also see

 * Parallelism is Reflexive Relation
 * Parallelism is Transitive Relation


 * Parallelism is Equivalence Relation