Q: Why does a circle belong in the quadrilateral family?

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No, the circle is inscribed in the quadrilateral.

It is an inscribed quadrilateral or cyclic quadrilateral.

you dont

cyclic

Of the shapes listed, only the rhombus is a quadrilateral.

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no

No, the circle is inscribed in the quadrilateral.

It is an inscribed quadrilateral or cyclic quadrilateral.

A quadrilateral has four sides. A circle does not have four sides. Therefore, a circle is not a 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.

No. You can have a very "thin" quadrilateral that is completely in the top half of the circumscribing circle. Then the centre of the circle will be below and OUSIDE the quadrilateral. The diagonals of the quadrialteral will be INSIDE the quadrilateral while they are within the circle and so cannot pass through the centre.

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

you dont

cyclic

no because it has an infinate number of sides a quadrilateral has exactly 4

Of the shapes listed, only the rhombus is a quadrilateral.

circle