How are they related?

The relation $\phi$ is just given to troll the problem solvers. It is just a normal relation.

Now, for reflexive relation, one line ahould be perpendicular to itself which is a fallacy. So, it is not reflexive. Thus, it is not an equivalence relation. Suppose line $a$ is perpendicular to line $b$ which in turn is perpendicular to another line $c$ , then line $a$ is not perpendicular but parallel to line $c$ . So it is not transitive.

Now, if line $a$ is perpendicular to line $b$ , then line $b$ is always perpendicular to $a$ . So, $\phi$ is a $\text{symmetric}$ relation.