Convex quadrilaterals are further classified as follows:
Trapezium (British English) or trapezoid (NAm.): two opposite sides are parallel.
Isosceles trapezium (Brit.) or isosceles trapezoid (NAm.): two opposite sides are parallel and the base angles are congruent. This implies that the other two sides are of equal length, and that the diagonals are of equal length. An alternative definition is a quadrilateral with an axis of symmetry bisecting one pair of opposite sides.
Trapezium (NAm.): no sides are parallel. (In British English this would be called an irregular quadrilateral, and was once called a trapezoid.)
Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other. A general term including square, rectangle, rhombus and rhomboid.
Kite: two adjacent sides are of equal length and the other two sides also of equal length. This implies that the angles between the two pairs of congruent sides are equal, and also implies that the diagonals are perpedicular. (It is common, especially in the discussions on plane tessellations, to refer to a concave kite as a dart or arrowhead.)
Rhombus or rhomb: all four sides are of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other. "A pushed-over square."
Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles). "A pushed-over rectangle."
Rectangle (sometimes referred to as an Oblong): all four angles are right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length.
Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. This implies that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle.
Rhombus (four equal sides) + Rectangle (four equal angles) = Square (four equal sides and four equal angles) → Parallelogram (opposite sides are parallel) → Quadrilateral (four-sided polygon)
Cyclic quadrilateral: the four vertices lie on a circumscribed circle.
Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
Bicentric quadrilateral: both cyclic and tangential.