In the first case,
the given figure has 2 sets of opposite sides congruent. Then, it is a parallelogram.
In the second case, each diagonal measure the same but they dont bisect each other.
Then, it is not necessarily a parallelogram.
In the third case, the vertical lines are parallel and the alternate interior angles are congruent.
This means that segments BD and AC are parallel too. Since a parallelogram has both pairs of opposite sides parallel, then this figure is a parallelogram.
In the fourth case, we have a pair of congruent angles:
however, we dont know something about the other pair of angles. Therefore, this figure is not necessarily a parallelogram
