A parallelogram is a quadrilateral (four-sided) polygon with two pairs of parallel sides.
A rhombus is a parallelogram with all 4 sides of equal measure.
A rectangle is a parallelogram with four vertices that are right angles. It is sufficient to prove that one vertex has a right angle because if it does, the other three have to be right angles as well. (In any parallelogram opposite angles are equal and adjacent angles are supplementary -- so if one angle is 90, the opposite is 90 and both adjacent angles are also 90 because 180 - 90 is 90.)
A square is either a rectangle with all 4 sides of equal measure, or a rhombus with 4 vertices that are right angles. Again, proof of one right angle is sufficient.
