Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.

(N/A) Let the two parallel lines be $l$ and $m$,such that $l \parallel m$. Let line $p$ be perpendicular to $l$ $(p \perp l)$ and line $n$ be perpendicular to $m$ $(n \perp m)$.
We need to show that $p \parallel n$.
Since $n \perp m$,the angle between them is $90^{\circ}$,so $\angle 1 = 90^{\circ}$.
Since $p \perp l$,the angle between them is $90^{\circ}$,so $\angle 2 = 90^{\circ}$.
Since $l \parallel m$ and $p$ acts as a transversal,the corresponding angles are equal,so $\angle 2 = \angle 3$.
Since $\angle 2 = 90^{\circ}$,it follows that $\angle 3 = 90^{\circ}$.
Now,considering lines $p$ and $n$ with transversal $m$,we have $\angle 1 = 90^{\circ}$ and $\angle 3 = 90^{\circ}$.
Since $\angle 1 = \angle 3$,these are corresponding angles,which implies that the lines $p$ and $n$ must be parallel.
Therefore,$p \parallel n$.