I think you want to prove that a rectangle is a quadrilateral. You can prove this with definitions. The definition of a quadrilateral is that it is a shape with four sides. Since a rectangle has four sides, that proves that a rectangle is a quadrilateral.
A kite has 4 sides and 4 angles.
Two angles are congruent and two angles are not.
The fact that two angles are congruent may help you.
I cannot see the picture to good