The quadrilaterals in the classification below display a side-angle symmetry, which is discussed more fully in my book Some Adventures in Euclidean Geometry, available as downloadable PDF or printed book.
All quadrilaterals are dynamic and can dragged into the special cases below them, which INHERIT ALL the properties of the linked general cases ABOVE them. Click on Definitions and some Properties of Quadrilaterals to open a separate, new window with a list of definitions and some important properties of these quadrilaterals.
Note: the right kite and isosceles circum trapezoid are both so-called bicentric quadrilaterals; in other words, subsets of those general quadrilaterals which are both cyclic and circumscribed (but not shown in the classification scheme). Bicentric quadrilaterals also have several interesting properties which can be found in books on advanced Euclidean geometry and mathematical journals. See Bicentric Quadrilateral Properties for some properties.
Sketchpad 4 and Sketchpad 5 sketches of this Hierarchical Classification Tree (with the exception of the Triangular Kite, Right Kite, Isosceles Circum Trapezoid and Trilateral Trapezoid) has Property buttons in the sketch, and can be downloaded for free from Sketchpad Exchange.
A suggested series of learning activities to visually introduce young children to the quadrilaterals and foster the idea of hierarchical class inclusion from the beginning is available at Classifying Quadrilaterals Visually.
Though definitions are to some extent arbitrary and we can often choose them as we wish in mathematics, sound pedagogical practice is NOT to just provide 'ready-made' definitions to students, but to involve students in actually defining and classifying the quadrilaterals themselves (e.g. see De Villiers, 1998). Sound pedagogical practice would also involve a classroom discussion and critical comparison of the merits and demerits of different possible definitions for the quadrilaterals, including carefully comparing hierarchical (inclusive) and partition (excusive) definitions (e.g. see De Villiers, 1994; De Villiers, 2009, pp. 13-28. and Jim King, undated.) A similar analysis and approach is advocated in the 2007 book The Classification of Quadrilaterals: A Study of Definition. by Zalman Usiskin and Jennifer Griffin, and is also highly recommended.
Michael de Villiers, Updated 22 April 2013.