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Properties of Quadrilaterals

Quadrilateral Classification

Definition Quadrilateral
A quadrilateral is a polygon with four sides.
Definition Parallelogram
A parallelogram is a quadrilateral with two pairs of opposite sides parallel.
Definition Square
A square is a quadrilateral with four right angles and four equal sides.
Definition Rectangle
A rectangle is a quadrilateral with four right angles.
Definition Rhombus
A rhombus is a quadrilateral with four equal sides.
Definition Trapezium
A trapezium is a quadrilateral with one pair of opposite sides parallel.

Properties

Proposition Properties of a Parallelogram
  • The opposite sides are equal in length.
  • The opposite angles are equal.
  • The adjacent angles are supplementary.
  • The diagonals bisect each other.
Proposition Properties of a Square
  • The opposite sides are parallel.
  • The diagonals bisect each other at right angles and are equal in length.
Proposition Properties of a Rectangle
  • The opposite sides are equal in length.
  • The opposite sides are parallel.
  • The diagonals bisect each other and are equal in length.
Proposition Properties of a Rhombus
  • The opposite sides are parallel.
  • The opposite angles are equal.
  • The adjacent angles are supplementary.
  • The diagonals bisect each other at right angles.

Angles

Proposition Sum of the Angles of a Quadrilateral
The sum of the angles of a quadrilateral is \(360^\circ\).

We divide the quadrilateral \(ABCD\) into two triangles, \(ABC\) and \(ACD\), using the diagonal \(AC\).
$$\begin{aligned}\text{Sum of the angles of quadrilateral } ABCD &= \text{Sum of angles of } \triangle ABC + \text{Sum of angles of } \triangle ACD \\ &= 180^\circ + 180^\circ \\ &= 360^\circ\end{aligned}$$

Example
Find the unknown angle \(x^\circ\).

The sum of the angles of a quadrilateral is \(360^\circ\). Given angles \(60^\circ\), \(95^\circ\), and \(80^\circ\):$$\begin{aligned}x^\circ + 95^\circ + 80^\circ + 60^\circ &= 360^\circ \\ x^\circ + 235^\circ &= 360^\circ \quad \text{(Adding known angles)} \\ x^\circ &= 360^\circ - 235^\circ \quad \text{(Subtracting 235 from both sides)} \\ x^\circ &= 125^\circ\end{aligned}$$