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Q: What is a quadrilateral inscribed in a circle called?

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No, the circle is inscribed in the quadrilateral.

The opposite angles of a quadrilateral inscribed in a circle have a sum of 180 degrees.

cyclic

yes

If a parallelogram is inscribed in a circle then it must be a cyclic quadrilateral.

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.

opposite angles are supplementary

There is no specific limitation on any one angle of an inscribed quadrilateral.

(99,90) (105,75)

An inscribed circle.

It is called incenter

No. For example, if one angle measures 100 degrees, and its adjacent angle is 80 degrees, then the opposite angles would be either 200 or 160 degrees, but in order for a quadrilateral to be inscribed in a circle the opposite angles would have to equal 180 degrees. A parallelogram can be inscribed in a circle if it is a rectangle.

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