If all four sides are equal, it must be a rhombus, which is a parallelogram. The square is a special case of a rhombus.
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It could be an arrowhead or a kite.
The answer is: quadrangle
That's not possible. No rhombus (quadrilateral with two pairs of equal sides) cannot be a parallelogram.
It could be an arrowhead.
A Kite. The two sets of equal sides are consecutive.