Answer:
A. If two coplanar lines do not intersect, then the lines are parallel. This is true. Two lines are parallel lines if they are coplanar and do not intersect.
B. Two lines lie in one plane and do not intersect if and only if the two lines are parallel. This is true. Only parallel lines lying in a plane do not intersect.
C. Two lines lie in one plane if and only if the lines are parallel. This is false. Two lines in the same 3 d space which do not intersect need not be parallel. They are parallel only if they lie in a common plane. If not, they are called skew lines.
D. If two lines are parallel, then they lie in one plane and do not intersect. Yes this is true.