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. .