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.

$\blacksquare$

Also see

• Perpendicularity is Antireflexive Relation
• Perpendicularity is Antitransitive Relation

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets