Definitions:  Parallelogram -- a quadrilateral that has two pairs of parallel sides.  Rectangle -- a quadrilateral with four right angles  Rhombus -- a quadrilateral with four congruent sides  Square -- a quadrilateral with four sides congruent and four right angles  Kite -- a quadrilateral with two distinct pairs of congruent consecutive sides.  Trapezoid -- a quadrilateral with at least one pair of parallel sides.  Isosceles trapezoid -- a quadrilateral with at least one pair of parallel sides in which the legs are congruent.

Parallelograms  If a quadrilateral is a parallelogram, then its opposite sides are congruent.  If a quadrilateral is a parallelogram, then its opposite angles are congruent.  If a quadrilateral is a parallelogram, then its diagonals bisect each other.  If a quadrilateral is a parallelogram, then consecutive angles are supplementary.  If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.  If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.  If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.  If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Rectangle  If a quadrilateral is a rectangle, then it is a parallelogram.  If a parallelogram is a rectangle, then its diagonals are congruent.  If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.  If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Rhombus  If a quadrilateral is a rhombus, then it is a parallelogram.  If a parallelogram is a rhombus, then its diagonals are perpendicular.  If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.  If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.  If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

Square  To prove a square, you must prove it is both a rectangle and a rhombus.

Kite  If a quadrilateral is a kite, then its diagonals are perpendicular.  If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.  If a quadrilateral is a kite, then one of the diagonals bisects the pair of non-congruent angles.  If a quadrilateral is a kite, then exactly one diagonal bisects the other.

Isosceles Trapezoid  If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.  If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.  A trapezoid is isosceles if and only if its diagonals are congruent.

Midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs.  The midsegment is parallel to each base and its length is one half the sum of the lengths of the bases.