Parallelism is Equivalence 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 an equivalence relation on $S$.

Proof
Checking in turn each of the criteria for equivalence:

Reflexivity
We have that for a straight line $l$:
 * $l \parallel l$

Thus $\parallel$ is seen to be reflexive.

Symmetry
If $l_1 \parallel l_2$ then $l_2 \parallel l_2$.

Thus $\parallel$ is seen to be symmetric.

Transitivity
From Parallelism is Transitive:


 * $l_1 \parallel l_2$ and $l_2 \parallel l_3$ implies $l_1 \parallel l_3$.

Thus $\parallel$ is seen to be transitive.

$\parallel$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.