If two planes are parallel, must all lines within those planes be parallel? Explain.

The lines within those planes do not necessarily be parallel with each other. The lines rather are coplanar but they can be drawn at different angles or directions which will make these lines intersect each other which makes them not parallel.

It may not be necessary because, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.