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Yes, the common name for an equiangular equilateral quadrilateral would be a square, a special instance of a rhombus (four sided figure with equal sides) which is a common interpretation of "diamond" in two dimensions. If by diamond you mean "kite" then no, by definition kites are not equiangular.
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What shape has four sides of equal of length but no right angles?
Its a diamond :]
What is a Shape with four equal sides and no right angles?
A diamond, a rhombus, or a parallelogram.
What is a diamond type of shape called in maths?
A quadrilateral with four equal-length sides and opposite angles equal (but not all four) is a rhombus. That's the usual description of a diamond shape.
What shape has two sets of parallel lines and all the sides are the equal and there are no right-angles?
That is a parallelogram; it could also be called a diamond shape.
Do all squares have two parallel sides?
All squares have TWO SETS of opposite, parallel sides. A square is a parallelogram with 4 equal sides and 4 equal angles (right angles) of which adjacent sides are perpendicular. The related parallelogram, the rhombus, or "diamond" shape, also has 4 equal sides, but no right angles, where opposite angles are equal and adjacent angles are supplementary (sum to 180 degrees).
