Check the picture below.
notice, it says that PT is perpendicular to SQ.
now onto the other one
well, a parallelogram with one right-angle and at least one pair of congruent sides, hell a rectangle is that, so is an square, so no dice.
a rhombus that has congruent diagonals, well, a square is that but so is a rectangle, no dice.
a parallelogram whose diagonals bisect each other, well, a square has that, sure, a rhombus too and a rectangle too, no dice.
a rectangle with perpendicular diagonals, well, a rhombus has perpendicular diagonals, however is not a rectangle, a square is a rectangle also with perpendicular diagonals too.