By definition, a trapezoid only must have exactly one pair of
parallel sides. An isosceles trapezoid does have one pair of
congruent sides, but not all trapezoids will have exactly one pair
of congruent sides.
The only real characteristic of a trapezoid is that one pair of
opposite sides is parallel. For an isosceles trapezoid, in addition
to one pair of opposite sides being parallel; the legs are
congruent; each pair of base angles is congruent; and the diagonals
Yes, but they would have to be the pair of sides which are not
parallel and, in that case, you will have an isosceles trapezium.
It is also possible for these two sides to be congruent with one
(but not both) of the remaining sides.