Yes it is possible, but is rarely used. These kites with a reflex angle are often called arrowheads, because they look like arrows.
Yes, but it usually isn't though of. It would be on the outside of each vertice.
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A reflex angle is an angle greater than 180 degrees but less than 360 degrees. A kite is a quadrilateral with two distinct pairs of adjacent sides that are congruent. The angles of a kite can vary, but a kite can have a reflex angle if the sum of the non-congruent angles is greater than 180 degrees.
Honey, a kite isn't out here doing geometry in the sky. A reflex angle is greater than 180 degrees, and a kite's angles are as acute as your ex's sense of commitment. So, no, a kite does not have a reflex angle, it's just flying high and minding its own business.