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Year 7 Interactive Maths - Second Edition


Quadrilaterals

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

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.

Parallelogram

Note:

AB is parallel to CD and AC is parallel to BD


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

Rectangle


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

Square


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.

Rhombus

Note:

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


Other Quadrilaterals

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.

Trapezium


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.

Kite


Irregular Quadrilateral

An irregular quadrilateral does not have any special properties.

Irregular Quadrilateral


Angle Sum of a Quadrilateral

The diagonal AC divides the quadrilateral into two triangles

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

So the angle sum of a quadrilateral is 360


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 10

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

Quadrilateral ABCD

Solution:

x = 60 as the angle sum of a quadrilateral is 360


Example 11

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

Find the value of the pronumerals x and y.

Solution:

x = 115 and y = 65 as the opposite angles of a parallelogram are equal in size.


Example 12

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

Find the value of the pronumerals x and y.

Solution:

y = 110 because the angles between two unequal sides of a kite are equal.  x = 75 as the angle sum of a quadrilateral is 360.


Key Terms

quadrilateral, diagonal, parallelogram, rectangle, square, rhombus, trapezium, regular trapezium, kite, irregular quadrilateral, angle sum of a quadrilateral


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