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A square is a parallelogram a parallelogram isn't always a square. The definition of parallelogram is a closed shape with two parallel sides. So a rectangle is a parallelogram and so is a square and a rombus. The definition of square is shape with four equal sides.
Diagonally opposite pairs of angles will always be equal, but the only time all angles in a quadrilateral parallelogram are equal is when that shape is a rectangle.
A shape that is a parallelogram but not a quadrilateral doesn't exist.
A shape must have 4 sides in order to be a parallelogram (minus other defining characteristics a shape must have to be a parallelogram).
Most kites are (its common). But there are other kites that are a different shape. The ones with a different shape are usually a bigger size.