Question

identify all pairs of parallel and perpendicular lines in the image below.
○ $overleftrightarrow{xy}perpoverleftrightarrow{ps}$, $overleftrightarrow{ps}paralleloverleftrightarrow{qt}$
○ $overleftrightarrow{xy}perpoverleftrightarrow{ps}$, $overleftrightarrow{xy}perpoverleftrightarrow{qt}$, $overleftrightarrow{xy}perpoverleftrightarrow{ru}$, $overleftrightarrow{ps}paralleloverleftrightarrow{qt}$
○ $overleftrightarrow{xy}perpoverleftrightarrow{ps}$, $overleftrightarrow{xy}perpoverleftrightarrow{qt}$, $overleftrightarrow{ps}paralleloverleftrightarrow{qt}$, $overleftrightarrow{ps}paralleloverleftrightarrow{ru}$
○ $overleftrightarrow{xy}perpoverleftrightarrow{ps}$, $overleftrightarrow{xy}perpoverleftrightarrow{qt}$, $overleftrightarrow{ps}paralleloverleftrightarrow{qt}$

Explanation:

Step1: Check perpendicular lines

Lines are perpendicular if they intersect at a right - angle. From the image, $\overleftrightarrow{XY}$ is perpendicular to $\overleftrightarrow{PS}$ and $\overleftrightarrow{XY}$ is perpendicular to $\overleftrightarrow{QT}$.

Step2: Check parallel lines

Parallel lines never intersect. Here, $\overleftrightarrow{PS}\parallel\overleftrightarrow{QT}$. There is no indication that $\overleftrightarrow{PS}\parallel\overleftrightarrow{RU}$ from the given image.

Answer:

$\overleftrightarrow{XY}\perp\overleftrightarrow{PS},\overleftrightarrow{XY}\perp\overleftrightarrow{QT},\overleftrightarrow{PS}\parallel\overleftrightarrow{QT}$ (the last option in the multiple - choice list)