No. A quadrilateral only must have four sides; those four sides can be arranged on any angle that permits the four sides to also have four angles.
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Yes, a parallelogram is always a quadrilateral.
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.
4, and the midsegments always form a parallelogram
No. A parallelogram is a special kind of quadrilateral.
A shape that is a parallelogram but not a quadrilateral doesn't exist.