Yes, both of these are special versions of quadrilateral. And a quadrilateral is a four sided polygon.
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No, a trapezoid is not a parallelogram. Irregular polygons are generally not parallel.
There are lot of shapes in geometry. It is called polygons. Triangle, trapezoid, circle, parallelogram, rhombus, rectangle, diagonal, square etc. Parallelogram, rhombus, rectangle, diagonal, square, and trapezoid are quadrilaterals.
Square = parallelogram and a square trapezoid = trapezoid Parallelogram = Parallelogram
trapezoid