A square, a rhombus or s kite would fit the given description
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A polygon with 2n sides (where n is an integer) which is symmetric about one diagonal.
A diagonal joining opposite angles is the principal diagonal. It may or may not bisect the angles, and that does not affect its name.
Not for every parallelogram. Only for a rhombus (diamond) or square will the diagonals bisect the opposite angles they connect, and diagonals are perpendicular. In rectangles, the diagonals do not bisect the angles and are notperpendicular, but they do bisect each other.
a rhombus
A square has 2 diagonals that are equal in length and bisect each other at right angles.
Yes, in the figure of a kite one diagonal bisects the other. They do not bisect each other.