The quadrilaterals below are listed in order from top to bottom, and from left to right in correspondence with the dynamic Hierarchical Quadrilateral Tree where this page is linked from, but the duals of each other are grouped together next to each other to display the sideangle duality. Only a selection of some important properties are given. Since quadrilaterals lower down in the hierarchy inherit ALL the properties from those they are linked to from above, only some additional properties are given lower down. Also note that each of the quadrilaterals below can be mathematically defined in several different, but equivalent ways. So the definitions below, mostly based on symmetry, are not unique.
Quadrilateral  Closed, plane figure with four vertices A, B, C and D, connected by four straight sides AB, BC, CD and DA. Properties: If convex, no reflexive angles, both diagonals interior & interior angle sum is 360^{0}. If concave, one reflexive angle, one diagonal exterior & interior angle sum is 360^{0}. If crossed, two reflexive angles, two diagonals exterior & interior angle sum is 720^{0}.
Circumscribed Quadrilateral  any quadrilateral circumscribed around a circle. Properties: 
Cyclic Quadrilateral  any quadrilateral inscribed in a circle. Properties: 
Perpendicular or Orthodiagonal Quadrilateral  any quadrilateral with perpendicular diagonals. Properties: 
Diagonal or Equidiagonal Quadrilateral  any quadrilateral with equal diagonals. Properties: 
Bisecting or Bisectdiagonal Quadrilateral  any quadrilateral with at least one of its diagonals bisected by the other. Properties: 
Trapezoid (or Trapezium)  any quadrilateral with at least one pair of opposite sides parallel. Properties: 
Kite  any quadrilateral with at least one axis of symmetry through a pair of opposite angles (vertices). Properties: 
Isosceles Trapezoid (or Trapezium)  any quadrilateral with at least one axis of symmetry through a pair of opposite sides. Properties: 
Parallelogram (selfdual)  any quadrilateral with halfturn (or point) symmetry. Properties: 

Rhombus  any quadrilateral with two axes of symmetry, each through a pair of opposite angles (vertices). Properties: 
Rectangle  any quadrilateral with two axes of symmetry, each through a pair of opposite sides. Properties: 
Triangular Kite  any kite with at least three equal angles. (In the sketch, these are at A, B and D). Properties: 
Trilateral Trapezoid (or Trapezium)  any isosceles trapezium with at least three equal sides. (In the sketch, these are AB, AD and DC). Properties: 
Right Kite  any kite inscribed in a circle. Properties: 
Isosceles Circum Trapezium  any isosceles trapezium circumscribed around a circle. Properties: 
Square (selfdual)  any rhombus with an axis of symmetry through a pair of opposite sides or any rectangle with an axis of symmetry through a pair of opposite angles (vertices). Properties: 
Note: Most of the quadrilateral properties above are proved in my book Some Adventures in Euclidean Geometry, and available as downloadable PDF or printed book. My Key Curriculum Press book Rethinking Proof with Sketchpad also contains some proof activities for an Isosceles Trapezoid, Cyclic Quadrilateral, a Circumscribed Quadrilateral, Rhombus, and the Interior Angle Sum of a Crossed Quadrilateral, and some discussion, and generalizations to higher polygons, in the Teacher Notes.
Michael de Villiers, created, 2008; most recently updated 26 September 2016.