Various Types of Quadrilaterals

Rectangle

A rectangle is a quadrilateral in which all four angles are right angles. Since the opposite angles in a rectangle are equal, it is a parallelogram.
Thus, in a rectangle, the opposite sides are equal in length, the opposite angles are equal, and the diagonals bisect each other.

[Explanation]

In the rectangle , drawing diagonals and , and considering and , we have:
From (1), (2), and (3), (by congruence), so
Hence, the diagonals of a rectangle are equal in length.

Properties of a Rectangle

The diagonals of a rectangle are equal in length and bisect each other.

Rhombus

A rhombus is a quadrilateral in which all four sides are equal in length. Since the opposite sides are equal, it is a parallelogram.
Thus, in a rhombus, the opposite sides and opposite angles are equal, and the diagonals bisect each other.

[Explanation]

In the rhombus , let the diagonals and intersect at . Considering and :
From (4), (5), and (6), (by congruence), so
Since , we have , which implies
Thus, the diagonals of a rhombus are perpendicular to each other.

Properties of a Rhombus

The diagonals of a rhombus are perpendicular and bisect each other.

Square


A square is a quadrilateral in which all four angles are equal and all four sides are equal. Hence, a square is both a rectangle and a rhombus.
Thus, in a square, the diagonals are equal in length and are perpendicular bisectors of each other.

Properties of a Square

The diagonals of a square are equal in length and are perpendicular bisectors of each other.

Properties of a Trapezoid

A trapezoid is a quadrilateral with one pair of opposite sides parallel. In particular, an isosceles trapezoid is one where the base angles are equal.

Relationships Between Different Types of Quadrilaterals

Among quadrilaterals:
  • If one pair of opposite sides is parallel, it is a trapezoid.
  • If both pairs of opposite sides are parallel, it is a parallelogram.
  • If one angle is a right angle, it becomes a rectangle.
  • If adjacent sides are equal, it becomes a rhombus.
  • If a rectangle has adjacent sides equal, it becomes a square.
  • If a rhombus has a right angle, it becomes a square.