Question

which statement is true regarding the parallel and perpendicular lines in the diagram?
( k parallel n ) and ( w perp m )
( k parallel n ) and ( n perp m )
( w parallel n ) and ( n perp m )
( w parallel n ) and ( w perp m )

Explanation:

Step1: Analyze parallel lines

Lines \(w\) and \(n\) are in the same plane \(\mathcal{H}\) and have the same direction (indicated by the arrowheads), so \(w\parallel n\).

Step2: Analyze perpendicular lines

Line \(m\) is perpendicular to the plane \(\mathcal{H}\), and since \(w\) is in plane \(\mathcal{H}\), by the property of a line perpendicular to a plane (a line perpendicular to a plane is perpendicular to every line in that plane), \(w\perp m\).

For the first option: \(k\) and \(n\) are not parallel ( \(k\) is not in the plane \(\mathcal{H}\) where \(n\) lies). For the second option: \(k\) and \(n\) are not parallel. For the third option: \(w\parallel n\) is correct, but \(n\perp m\) is wrong ( \(m\) is perpendicular to the plane containing \(n\), not \(n\) itself in the sense of two - dimensional perpendicularity within a plane for \(n\) and \(m\) as presented in the diagram). So the correct statement is \(w \parallel n\) and \(w\perp m\).

Answer:

\(w \parallel n\) and \(w\perp m\)