Some of these special quadrilaterals include others. In this sense, they "inherit" their properties.  For example, all rectangles are parallelograms, and all squares are rectangles (and therefore also parallelograms).  The following flowchart should help keep this straight.  Each object is also an object it is connected to above and has all the properties of the quadrilateral above:

For example, a rhombus is both a kite and a parallelogram, and since a parallelogram is an isosceles trapezoid, a rhombus is also an isosceles trapezoid, a trapezoid, and of course a quadrilateral.