opposite angles are supplementary
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That is the definition of the incenter; it is the center of the inscribed circle.
Yes. The corners must be right angles for it to be inscribed on the circle.
not always, since sometimes kites may have a pair of opposite angles that are not supplementary. Special types of kites can be inscribed.
No, they are supplementary.
An inscribed angle is an angle with its vertex on a circle and with sides that contain chords of the circle.