sure... it just needs four sides
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I call it a [non-isosceles] trapezium, but those on the other side of the pond call it a [non-isosceles] trapezoid. If no sides are equal, it is called scalene trapezoid
Symmetric
No.
It is an inscribed quadrilateral or cyclic quadrilateral.
"It is a quadrilateral with exactly one pair of parallel sides." is the true one. Some people define it as having at least one pair of parallel sides, so if your teacher is one of those people then all of the answers are wrong, but since all of the other answers are definitely wrong then this is the only one that could be right.