Then it's consecutive angles are supplementary.
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An equilateral quadrilateral which would include a square.
In a parallelogram consecutive angles are always supplementary. This means they equal 180.
It is a parallelogram or a rhombus
Yes.
Not necessarily. A parallelogram is a quadrilateral with 2 sets of 2 parallel sides. Only rectangles and squares have right angles as all four angles - a rectangle is a "special" parallelogram that has all four angles equal; a square is a "special" rectangle that has all four sides of equal length.