A symmetrical trapezium is specified by the lengths of the base and the height and the angle of inclination of the non‐parallel sides to the base. The internal bisectors of the angles of the trapezium are used to define a kite and this first kite is used to define a second kite. The properties of these kites are developed for two distinct configurations. The trapezium is allowed to vary and the conditions for the kites to be enclosed by the trapezia are investigated using inequalities. Special limiting cases and an alternative specification are considered.