1. Quadrilaterals MATH 124

2. Quadrilaterals All quadrilaterals have four sides. All sides are line segments that connect at endpoints. The most widely accepted definition of quadrilateral doesn’t allow sides to cross. The sum of all interior angles in a quadrilateral is 360 degrees, and the sum of all exterior angles in a quadrilateral is also 360 degrees.

3. Trapezoid, first definition A trapezoid is a quadrilateral with at least one pair of parallel sides. These sides are called bases. A general trapezoid doesn’t have any special properties. All sides generally have different lengths, diagonals are of different lengths, and no angles are congruent. Note that adjacent angles, those lying on different bases, are supplementary. Under this definition of trapezoid, all parallelograms are trapezoids. The only kite that is also a trapezoid is the rhombus.

4. Trapezoid, second definition A trapezoid is a quadrilateral with exactly one pair of parallel sides. Again, a general trapezoid does not have any special properties. Under this definition, a trapezoid can never be a kite or a parallelogram. While this definition is more commonly used in U.S. textbooks, it’s also a lot more restrictive.

5. Isosceles trapezoid Under the first definition, an isosceles trapezoid is a trapezoid with at least one pair of congruent sides. Under the second definition, an isosceles trapezoid is a trapezoid with exactly one pair of congruent sides. These are the two sides that are not bases. An isosceles trapezoid has one line of symmetry, which goes through the midpoints of the bases, BUT ONLY is the trapezoid is not a parallelogram. The diagonals of an isosceles trapezoid are congruent, but they do not bisect each other or angles and are generally not perpendicular. The base angles in an isosceles trapezoid are congruent BUT ONLY if the trapezoid is not a parallelogram.

6. Kite A kite is a quadrilateral with two pairs of adjacent congruent sides. It has one pair of congruent angles, the ones formed by sides of different lengths. The diagonal that forms a triangle with two sides of different lengths, also called the main diagonal, is a line of symmetry for the kite. The main diagonal bisects the angles it goes through. The main diagonal bisects the other diagonal. The diagonals are perpendicular. A rhombus is always a kite, and a kite is sometimes a rhombus. A parallelogram is a kite only if it is a rhombus. A trapezoid is a kite only if it is a rhombus.

7. Parallelogram A parallelogram is a quadrilateral with two pairs of opposite parallel sides. The opposite sides in a parallelogram are congruent. The diagonals are not congruent, but they do bisect each other. They are generally not perpendicular, and they do not bisect angles. The opposite angles are congruent, and the adjacent angles are supplementary. A parallelogram generally does not have line symmetry, but does have 180 degree rotation symmetry. The only parallelogram that is also a kite is a rhombus. All parallelograms are trapezoids by the first definition.

8. Rhombus A rhombus is a parallelogram whose all sides are congruent. As a parallelogram, it has all the properties of parallelograms (parallel opposite sides, supplementary adjacent angles, congruent opposite angles). The diagonals or a rhombus are not congruent, but do bisect each other and the angles. They are also perpendicular. The diagonals of a rhombus are its lines of symmetry. A rhombus is also a kite and a trapezoid (under the first definition).

9. Rectangle A rectangle is a parallelogram with all congruent angles. The angles all measure 90 degrees. As a parallelogram, it has all the properties of parallelograms (parallel and congruent opposite sides). The diagonals of a rectangle are congruent. They bisect each other, but not the angles, and they are not perpendicular. A rectangle has two lines of symmetry, which pass through midpoints of opposite sides. It also has 180 degree rotation symmetry.

10. Square A square is a regular quadrilateral. It is equilateral and equiangular. This means that it is a rhombus and rectangle at the same time. As a parallelogram, it has the properties of the other parallelograms, which include parallel opposite sides. A square has congruent diagonals, which bisect each other at a 90 degree angle, and also bisect angles. A square has four lines of symmetry: the diagonals and lines connecting midpoints of opposite sides. It also has 90 degree rotation symmetry. A square belongs to all the categories already described.