Question

are the lines in the diagram perpendicular, parallel, skew, or none of these?
l and m:
l and n:
m and n:

Explanation:

for \( l \) and \( m \):

Step1: Analyze line directions

Lines \( l \) and \( m \) lie on plane \( A \), and their directions are such that they never meet (parallel) and have the same slope (or direction vector relation).

Step2: Conclude relationship

Since they are coplanar and never intersect, \( l \) and \( m \) are parallel.

for \( l \) and \( n \):

Step1: Check plane and angle

Line \( l \) is in plane \( A \), line \( n \) is in plane \( B \). The diagram shows a right angle (the orange square) at their intersection, meaning the angle between them is \( 90^\circ \).

Step2: Conclude relationship

If two lines intersect at \( 90^\circ \), they are perpendicular. So \( l \) and \( n \) are perpendicular.

for \( m \) and \( n \):

Step1: Check plane and angle

Line \( m \) is in plane \( A \), line \( n \) is in plane \( B \). The diagram shows a right angle (the orange square) at their intersection, meaning the angle between them is \( 90^\circ \).

Step2: Conclude relationship

If two lines intersect at \( 90^\circ \), they are perpendicular. So \( m \) and \( n \) are perpendicular.

Answer:

\( l \) and \( m \): parallel
\( l \) and \( n \): perpendicular
\( m \) and \( n \): perpendicular