There are infinitely many polygons which have this property. The easiest to think of is a right triangle. But there are also quadrilaterals, pentagons, etc., that can be constructed that have this property.
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shape no pairs of perpendicular sides
Trapezium
No. By definition, a rectangle must have TWO pairs of parallel sides. Also, each pair of parallel sides will be congruent. *If this is not happening with your shape, it isn't a rectangle.
A trapezium or parallelogram. A pentagon can have a pair of parallel sides. Polygons with 6 or more sides can have two (or more) pairs of parallel sides.
A trapezoid.