Prove that two lines that are respectively perpendicular to two intersecting lines intersect each other.

(N/A) Let $l$ and $m$ be two intersecting lines. Let line $n$ be perpendicular to $l$ $(n \perp l)$ and line $p$ be perpendicular to $m$ $(p \perp m)$.
To prove: Lines $n$ and $p$ intersect each other.
Proof: Assume for the sake of contradiction that lines $n$ and $p$ are parallel to each other $(n \parallel p)$.
Since $n \perp l$ and $n \parallel p$,it follows that $p \perp l$ (because lines perpendicular to the same line are parallel,or conversely,a line perpendicular to one of two parallel lines is perpendicular to the other).
We are given that $p \perp m$. Thus,we have $p \perp l$ and $p \perp m$.
This implies that $l \parallel m$ (since both are perpendicular to the same line $p$).
However,this contradicts the given information that $l$ and $m$ are intersecting lines.
Therefore,our assumption that $n \parallel p$ must be false.
Hence,lines $n$ and $p$ must intersect each other.