A square seems to fit the given description if it also has 4 equal sides
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Well, honey, that sounds like a rectangle to me. It's got those four right angles, two diagonals for days, and don't forget those four axes of symmetry. So, if you're looking for a shape that fits that description, you've hit the jackpot with a good ol' rectangle.
A rectangle. Obviously the right angles are in the four corners of the rectangle. The lines of symmetry occur across the horizontal and vertical. There are no lines of symmetry on the diagonal.
octagon
A parallelogram
A trapezoid Trapezoid - 2 congruent diagonals that do not bisect each other. No right angles and has 1 pair of opposite parallel sides.