Kites in Taxicab Geometry
Kites in Euclidean, Hyperbolic, and Spherical Geometries
Kites in Euclidean, Hyperbolic, and Spherical Geometries all have similar characteristics arising from the fact that the major diagonal divides the kite into two triangles that are congruent with a reflection. The interior angles at the ends of the minor diagonal are congruent. The major diagonal is an angle bisector on both ends, and the minor diagonal is perpendicularly bisected by the line containing the major diagonal.
General Kites in Taxicab Geometry
General Kites in Taxicab Geometry share none of the properties of kites in Euclidean Geometry, other than the defining characteristic of two disjoint pair of congruent consecutive sides.
Taxicab Kites with Reflective Symmetry
If the major diagonal of a Taxicab kite is vertical or horizontal, or makes an angle of 45 degrees or -45 degrees with the vertical or horizontal, then the Taxicab Kite is also a Euclidean Kite, and it has all the normal properties of Euclidean Kites.
So, in Euclidean, Hyperbolic, and Spherical Geometries the class of quadrilaterals with reflective symmetry is identical with the class of kites. In Taxicab Geometry the class of quadrilaterals with reflective symmetry is a proper subset of the class of kites.