Review: properties of quadrilaterals

key notes :

Definition of Quadrilateral

  • A quadrilateral is a four-sided polygon with four angles and four vertices.
  • The sum of the interior angles of a quadrilateral is always 360°.

Types of Quadrilaterals and Their Properties

Parallelogram

  • Opposite sides are parallel and equal.
  • Opposite angles are equal.
  • Diagonals bisect each other.

Rectangle

  • All angles are 90°.
  • Opposite sides are equal and parallel.
  • Diagonals are equal and bisect each other.

Square

  • All sides are equal and opposite sides are parallel.
  • All angles are 90°.
  • Diagonals are equal, bisect each other, and are perpendicular.

Rhombus

  • All sides are equal.
  • Opposite sides are parallel, and opposite angles are equal.
  • Diagonals bisect each other at right angles (90°).

Trapezium (Trapezoid in US)

  • Only one pair of opposite sides is parallel.
  • Non-parallel sides are called legs.
  • If legs are equal, it is called an isosceles trapezium, where base angles are equal and diagonals are equal.

Kite

  • Two pairs of adjacent sides are equal.
  • One pair of opposite angles is equal.
  • Diagonals intersect at right angles, and one diagonal bisects the other.

Special Properties of Diagonals in Quadrilaterals

  • Parallelogram: Diagonals bisect each other.
  • Rectangle: Diagonals are equal and bisect each other.
  • Square: Diagonals are equal, bisect each other, and are perpendicular.
  • Rhombus: Diagonals bisect each other at right angles.
  • Kite: One diagonal is bisected by the other at right angles.

Angle Sum Property

  • The sum of all interior angles of any quadrilateral is 360°.
  • The sum of exterior angles is always 360°, with each exterior angle formed by extending a side.

Midpoint Theorem for Quadrilaterals

  • The midpoints of the sides of any quadrilateral form a parallelogram.

Area Formulas of Quadrilaterals

  • Parallelogram: Base × Height
  • Rectangle: Length × Breadth
  • Square: Side2
  • Rhombus: 1/2 × Diagonal1 × Diagonal2
  • Trapezium: 1/2 × (Base1 + Base2) × Height
  • Kite: 1/2 × Diagonal1 × Diagonal2

Learn with an example

Is parallelogram GHIJ a rhombus?

  • yes
  • no

Since ∠GHJ ≅ ∠GJH ≅ ∠HJI ≅ ∠IHJ, HJ bisects ∠GHI and ∠GJI.

So, GHIJ is a rhombus.

Is parallelogram DEFG a rhombus?

  • yes
  • no

Since ∠EG F= ∠DEG = 46° and ∠DGE = ∠FEG = 48°, EG does not bisect ∠DGF and ∠DEF.

So, DEFG is not a rhombus.

Is parallelogram WXYZ a rhombus?

  • yes
  • no

Since ∠YVZ = 88°, WY and XZ are not perpendicular.

So, WXYZ is not a rhombus.

Let’s practice!