a quadrilateral has 4 (quad) sides. a square is a quadrilateral.
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A quadrilateral is inscribed in a circle it means all the vertices of quadrilateral are touching the circle. therefore it is a cyclic quadrilateral and sum of the opposite angles in cyclic quadrilateral is supplementary. suppose if one angle is A then another will be 180 degree - angle A.
Quadrilaterals usually have special names only if they DO have parallel sides, or some other special property. You might just call it "quadrilateral", or "a quadrilateral with no parallel sides".
Some trapezoids, not all.
Yes.
It can, but they can also all be the same.