No. If it did, it would be a square.
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A cyclic kite. A kite is cyclic if it has only two right angles.
There is no name for such a figure. Quadrilaterals such as a trapezium or kite or even one with no equal sides can have a right angle.
Yes
If ONLY two right angles, then it could be a right angled trapezium or a kite If more than two it must be a rectangle (a square is an equilateral rectangle).
Yes because the diagonals are perpendicular producing 4 right angles