How about a trapezoid as an example.
The Opposite Sides Parallel and Congruent Theorem states that if a quadrilateral has a pair of opposite sides that are parallel and congruent, then the quadrilateral is a parallelogram.
Then, the opposite sides are congruent.
No. The congruent sides need not be parallel.
sometimes
Parallel
No, it is not.
A kite
An isosceles trapezium
No, you need both pairs of opposite sides to be congruent (i.e. the same size) to be a parallelogram.
rhombus
square
Yes, it is.