No. Equilateral simply means equal sided. The shape need not have 4 sides and so need not be a parallelogram.
If the shape is a quadrilateral, and has either four equal sides or two sets of equal opposite sides, then it is a parallelogram.
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No, rhombus refers to its shape, not the fact that is has four equal sides. You could call a rhombus an equilateral parallelogram by why would you want to? Also, would you then call a square a rhombic rectangle?
Yes, but only if it is a square. A square is an equiangular rhombus, and a rhombus is an equilateral parallelogram.
Rhombus.
A Rhombus.
A parallelogram with 4 equal sides is a rhombus