If $\alpha ,\beta$ be straight lines in a plane, then check $R_{1}and{R}_2$ for being reflexive, symmetric and transitive $\alpha R_1\beta if\alpha \perp \beta and\alpha R_2\beta if\alpha \mid \mid \beta .$

$\alpha R_1\beta if\alpha \perp \beta$
where $\alpha ,\beta$ are straight lines in a plane.
Let $L$ be set of lines in a plane

Reflexivity:
No line is perpendicular to itself.
i.e.$\alpha$ is not perpendicular to $\alpha$    $\forall \alpha \in L$
So, $\alpha$ is not related to $\alpha$    $\forall \alpha \in L$
Hence, $R_1$ is not reflexive.

Symmetry:
Let $\alpha R_1\beta$
$\Rightarrow \beta R_1\alpha$       (Two lines are mutually perpendicular to each other)
Hence, $R_1$ is symmetric

Transitivity:
Let $\alpha ,\beta ,\gamma \in L$
Let $\alpha R_1\beta$ ; $\beta R_1\gamma$
It does not implies that $\alpha \perp \gamma$
Hence, $\alpha$ not related to $\gamma$
Hence, $R_1$ is not transitive.

Given $\alpha R_2\beta if\alpha \mid \mid \beta$
Reflexivity: 
Every line is parallel to itself.
$\alpha R_2\alpha$
Hence, $R_2$ is reflexive.

Symmetry:
Let $\alpha R_2\beta$
$\Rightarrow \alpha \parallel \beta$
$\Rightarrow \beta \parallel \alpha$ 
$\Rightarrow \beta R_2\alpha$
Hence, $R_2$ is symmetric.

Transitivity:
Let $\alpha R_2\beta ,\beta R_2\gamma$
$\Rightarrow \alpha \parallel \beta ;\beta \parallel \gamma$     (Two lines parallel to same line are parallel to each other.)
$\Rightarrow \alpha \parallel \gamma$
Hence, $R_2$ is transitive.