Quite simply - No.
Quadruped - a four legged animal, like a horse
Quad-bike - a four-wheeled all-terrain vehicle
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That could be any quadrilateral or pentagon.
an isosceles trapeziod could be a quadrilateral because it has 4 sides and a quadrilateral has 4 sides
It is A because a quadrilateral has 4 sides
It cannot exist. By definition, a quadrilateral only has 4 sides (from the word "quad" meaning 4)
A quadrilateral is a polygon with 4 sides... so if it has 4 sides it is a quadrilateral. your question doesnt make sense because a 4 sided quadrilateral could be any quadrilateral at all.