What is a parallelogram with all sides congruent called?

A parallelogram with all sides congruent is called a rhombus. All the sides of a rhombus are equal in length, which is why it is also called an equilateral quadrilateral. Other names commonly used to refer to a rhombus are "lozenge" and "diamond."

A rhombus has equal opposite angles; two of its diagonals are perpendicular, and its diagonals bisect opposite angles. A quadrilateral is a rhombus only if all sides of the quadrilateral are of equal length or if each diagonal bisects two opposite interior angles. A rhombus is a parallelogram in which the diagonals are perpendicular or at least two consecutive sides are equal in length.