Quadrilaterals and their properties

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What is a quadrilateral? 

A quadrilateral is a polygon with 4 sides and 4 vertices.

Types of quadrilaterals 

Square

Properties of a square

A square has 

  • four equal sides - \(\text {Side } AB = \text {Side } BC = \text {Side } CD = \text {Side } AD\)
  • four right angles - \(\begin{align} \angle A =  \angle B =  \angle C =  \angle D = \,90^\circ \end{align}\)
  • two pairs of parallel sides - \(\begin{align} AB ∥ DC \qquad  AD ∥ BC \end{align}\)

Rectangle

Properties of a rectangle 

A rectangle has 

  • two pairs of parallel sides - \(\begin{align} AB ∥ DC \qquad AD ∥ BC \end{align}\)
  • four right angles - \(\begin{align} \angle A =  \angle B =  \angle C =  \angle D = \,90^\circ \end{align}\)
  • opposite sides of equal length - \(AD = BC\) & \(AB = DC\)

Parallelogram    

Properties of a parallelogram

  • two pairs of parallel sides \(\begin{align}PQ ∥ RT \qquad PR ∥ QT \end{align}\)
  • opposite sides are of equal length. \(\text{Side }PQ = \text{Side }RT\)  and  \(\text{Side }PR = \text{Side }QT\)

Trapezium

Properties of a trapezium

  • One pair of parallel sides. \(\begin{align} EH ∥ GH \end{align}\)

Rhombus

A rhombus has

  • two pairs of parallel sides.  \(\begin{align} EH ∥ FG \qquad \qquad  EF ∥ HG \end{align}\)
  • four equal sides. \(\text{Side }EH = \text{Side }HG = \text{Side }GF = \text{Side }FE\)
  • Opposite angles are equal. \(\begin{align} \angle H = \angle F = \angle E = \angle G \end{align}\)

Angle sum property of quadrilaterals

The sum of the angles of a quadrilateral is always \(360^\circ\). This property is referred to as the angle sum property of a quadrilateral.

In the figure given above the sum of angles \(A\), \(B\), \(C\) & \(D\) is \(360^\circ\). This property comes handy when calculating unknown measure of angles in a quadrilateral.

Tips and Tricks

  • Tip: While naming a quadrilateral you can name it in anti-clockwise direction. For example, in the figure given below the quadrilateral can be named as \(\square \,ABCD\) and \(\square \,ACBD\).

  • While calculating the measures of an unknown angles in a quadrilateral dividing the quadrilateral into two triangles would help.

Common mistakes or misconceptions

Misconception 1: Any four sided figure is a quadrilateral. For example, the figure given below is a quadrilateral.

Though the above given has 4 sides and is a closed figure, it is not a quadrilateral. Quadrilaterals are closed figures made of non-intersecting line segments.

Misconception 2: All rectangles are squares
Many rectangles do not have 4 equal sides and 4 equal angles. However, all squares are rectangles as they all have 4 right angles, their opposite sides are parallel to each other and opposite sides are equal to each other.

Misconception 3: All rhombuses are squares
Many rhombuses do not have four equal sides and four equal angles. However, all squares are rhombuses as it has 4 equal sides, its opposite sides are parallel to each other.

Test your knowledge

Identify and name the quadrilaterals.

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This is ______________________

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This is ______________________

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This is ______________________

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This is ______________________

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This is ______________________

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