No. A parallelogram has two pairs of opposite angles that are equal but to be equiangular, all four angles must be the same. That is not necessarily the case in a parallelogram.
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Yes, if it is an equiangular or equiangular and equilateral parallelogram (rectangle or square).
The rectangle.
square or rhombus
A rhombus cannot be a square. A rectangle is an equiangular parallelogram. A rhombus is an equilateral parallelogram. A square is an equilateral and equiangular parallelogram. Thus, a square is both a rectangle and a rhombus.
Yes, a regular polygon is always equiangular.