Question

planes q and r are parallel planes. plane q contains line a. plane r contains line b.
if a third plane could be drawn which contains both lines a and b, then
lines a and b must be parallel.
lines a and b cannot be parallel.
lines a and b must be skew.
lines a and b must be perpendicular.

Explanation:

1. Recall the properties of parallel planes and lines in space:

• Parallel planes (Q and R) do not intersect.
• If a third plane contains both lines \(a\) (in Q) and \(b\) (in R), then lines \(a\) and \(b\) lie in a common plane (the third plane).
• Skew lines are non - parallel and non - intersecting lines that do not lie in the same plane. Since \(a\) and \(b\) lie in a common plane, they cannot be skew.
• For two lines in a plane, if the planes containing them (Q and R) are parallel, and they lie in a third plane, the only way this is possible is if the lines are parallel. If they were not parallel, they would intersect (since they are in a plane), but the original planes Q and R are parallel and do not intersect, so the lines \(a\) and \(b\) (in Q and R respectively) can only intersect if the planes intersect, which they don't. So lines \(a\) and \(b\) must be parallel. The option that says they cannot be parallel is wrong. The option that says they must be skew is wrong because skew lines don't lie in a plane. The option that says they must be perpendicular is wrong because there's no requirement for them to be perpendicular, they just need to be parallel.

Answer:

A. lines a and b must be parallel.