A kite formed with two congruent right triangles is called a right kite. In such a kite, two equal angles on a opposite sides of symmetry axis are each 90°. You can inscribe and circumscribe circles on a right kite. But the centers of the two circles are different(O₁ and O₂ as shown). As such it also called a bicentric quadrilateral.

A kite which has diagonals of equal lengths is called a equidiagonal kite. The angles in such kite are 60°, 75°, 150° and 75°.

Note: A "kite" is also a common name for certain birds of prey.