Perpendicularity is Symmetric Relation

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

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

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

Proof
Let $l_1 \perp l_2$.

By definition of perpendicular lines, $l_1$ meets $l_2$ at a right angle.

Hence $l_2$ similarly meets $l_1$ at a right angle.

That is:
 * $l_2 \perp l_1$

Thus $\parallel$ is seen to be symmetric.

Also see

 * Perpendicularity is Antireflexive Relation
 * Perpendicularity is Antitransitive Relation