Properties of parallelograms and special quadrilaterals

A parallelogram has both pairs of opposite sides parallel, and it follows that its opposite sides are congruent, its opposite angles are congruent, its consecutive angles are supplementary, and its diagonals bisect each other. You can prove a quadrilateral is a parallelogram by showing any of these: both pairs of opposite sides are congruent, both pairs of opposite angles are congruent, the diagonals bisect each other, or one pair of opposite sides is both parallel and congruent. Special parallelograms add more: a rectangle has four right angles and congruent diagonals; a rhombus has four congruent sides and perpendicular diagonals that bisect its angles; and a square is both a rectangle and a rhombus, so it has all of these properties. Outside the parallelogram family, a trapezoid has a pair of parallel sides (an isosceles trapezoid also has congruent legs, base angles, and diagonals), and a kite has two pairs of congruent adjacent sides and perpendicular diagonals. The special cases can be proved too: a parallelogram with congruent diagonals or one right angle is a rectangle, a parallelogram with perpendicular diagonals, two congruent adjacent sides, or diagonals that bisect its angles is a rhombus, and a parallelogram that is both a rectangle and a rhombus is a square.

This explanation is AI-generated and can occasionally be wrong. 🚩 Report