Square, rectangle, rhombus, parallelogram, kite and trapezoid are all special names for different types of quadrilaterals.
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Yes- a square is a special kind of quadrilateral.
quadrilateral
Most quadrilaterals do not have opposite sides which are equal in length. Sadly, there is no special name for them.
A trapezoid is a special case of a quadrilateral which has 1 pair of parallel sides.