Year 8 Interactive Maths - Second Edition

A quadrilateral is a 2-dimensional closed shape with four straight sides.  E.g. The shape ABCD shown here is a quadrilateral.

A line segment drawn from one vertex of a quadrilateral to the opposite vertex is called a diagonal of the quadrilateral.  AC is a diagonal of quadrilateral ABCD, as is BD.

## Types of Quadrilaterals

There are seven types of quadrilaterals that can be divided into two groups:  parallelograms and other quadrilaterals.

### Parallelograms

Quadrilaterals are called parallelograms if both pairs of opposite sides are equal and parallel to each other.  Different parallelograms and their properties are described below.

###### Parallelogram
• Opposite sides of a parallelogram are parallel and equal in length.
• Opposite angles are equal in size.

###### Rectangle
• Opposite sides of a rectangle are parallel and equal in length.
• All angles are equal to 90°.

###### Square
• Opposite sides of a square are parallel and all sides are equal in length.
• All angles are equal to 90°.

###### Rhombus
• All sides of a rhombus are equal in length
• Opposite sides are parallel.
• Opposite angles of a rhombus are equal.
• The diagonals of a rhombus bisect each other at right angles.

###### Note:

Rectangles, squares and rhombuses (or diamonds) are parallelograms.

Other quadrilaterals include trapeziums, kites and irregular quadrilaterals.

###### Trapezium
• A trapezium has one pair of opposite sides parallel.
• A regular trapezium has non-parallel sides equal and its base angles are equal, as shown in the diagram.

###### Kite
• Two pairs of adjacent sides of a kite are equal in length
• One pair of opposite angles (the ones that are between the sides of unequal length) are equal in size.
• One diagonal bisects the other.
• Diagonals intersect at right angles.

An irregular quadrilateral does not have any special properties.

## Angle Sum of a Quadrilateral

Draw a quadrilateral ABCD.  Then draw the diagonal AC to form two triangles.

### Finding the Fourth Angle of a Quadrilateral

If the measurements of three angles of a quadrilateral are known, then the missing angle can be calculated.

#### Example 21

Find the value of the pronumeral x in the accompanying diagram.  Give reasons for your answer.

#### Example 22

Find the value of the pronumerals in the following diagram.  Give reasons for your answer.

#### Example 23

Find the value of the pronumerals in the following diagram.  Give reasons for your answer.