It is not easy to show, but they certainly exist. For example, consider the quadrilateral with vertices at (0,0), (1,0), (0,1) and any point in the first quadrant other than (1,1)
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To show a right angle you have to mark the inside corner so that it looks like a square.
If you wan't to show that a pentagon has no right angles, then don't put any right angle signs in the figure.
a square, rectangle, rhombus.....
That's exactly what the box means! That box is there to show that an angle is right, as right angles can have different properties than acute or obtuse angles.
To prove that a quadrilateral is a right trapezoid, you need to show that it has one pair of parallel sides and one pair of right angles. This can be done by using the properties of parallel lines and perpendicular lines.