It is a trapezoid that can have two right angles plus one obtuse angle plus one acute angle and the four angles add up to 360 degrees.
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It is called a Rhombus.
A quadrilateral with 4 right angles can only be a rectangle or a square. A rectangle has only two lines of symmetry - the lines joining the midpoints of its opposite sides. So the answer cannot be a rectangle. A square has the same lines of symmetry as a rectangle, plus the two diagonals - 4 lines in all.
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 right angle trapezoid
This is a parallelogram.
A Rhombus.
Parrallelogram (it has rotational symmetry but no lines of symmetry)
square
A trapezoid is a quadrilateral with a set of parallel lines. Because it is a quadrilateral, it has four angles.
A trapezoid.
Parallelogram
rectangle
A square has.
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trapezoid
Such a shape cannot exist.