The description given fits that of a trapezoid which has a pair of opposite parallel sides of different lengths and 4 interior angles that add up to 360 degrees
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a trapezoid
I don't think this is possible. there is not a quadrilateral with these qualities. If it is a quadrilateral, it will automatically have parallel lines, but there is not one that has both. The closest one would be the trapezoid, with a set of parallel lines, but no right angle. The square and rectangle have two sets of parallel lines and 4 right angles.
A square has 4 equal sides, 2 pairs of parallel sides and 4 right angles.
A rectangular prism has 3 sets of 4 parallel lines and 24 angles.
No shape does. If it has 4 sides and one pair of parallel lines it cannot have only 1 right angle, it must either have: 1) 2 right angles, 1 acute acute and 1 obtuse angle; or 2) no right angles, 2 acute angles and 2 obtuse angles; in which case it would be a trapezium (trapezoid).