G S Rehill's Interactive Maths Software Series - "Building a Strong Foundation in Mathematics" from mathsteacher.com.au.

Order a 12-month Year 7 Interactive Maths software Homework Licence for only $19.95. Order a 12-month Year 8 Interactive Maths software Homework Licence for only $19.95. Order a 12-month Year 9 Interactive Maths software Homework Licence for only $19.95. Order a 12-month Year 10 Interactive Maths software Homework Licence for only $19.95.

Year 10 Interactive Maths - Second Edition


Quadrilaterals

A quadrilateral is a closed plane figure bounded by four line segments. For example, the figure ABCD shown here is a quadrilateral.

Quadrilateral ABCD has two diagonals in AC and BD.

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


Types of Quadrilaterals and their Properties

There are six basic types of quadrilaterals:

1.  Rectangle
  • Opposite sides are parallel and equal.
  • All angles are 90.
  • The diagonals bisect each other.

A rectangle.


2.  Square
  • Opposite sides are parallel and all sides are equal.
  • All angles are 90.
  • Diagonals bisect each other at right angles.

A square.


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

A parallelogram.


4.  Rhombus
  • All sides are equal and opposite sides are parallel.
  • Opposite angles are equal.
  • The diagonals bisect each other at right angles.

A rhombus.


5.  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 following diagram.

A trapezium.


6.  Kite
  • Two pairs of adjacent sides are equal.
  • One pair of opposite angles is equal.
  • Diagonals intersect at right angles.
  • The longest diagonal bisects the shortest diagonal into two equal parts.

A kite.


Theorem 3

Prove that the angle sum of a quadrilateral is equal to 360.

Proof:

A diagonal AC divides the quadrilateral ABCD into two triangles.  q and v are two angles in triangle ACD and p and u are two angles in ABC.

In triangle ABC, p + u + B = 180 degrees   {Angle sum of triangle}   ...(1).  In triangle ACD, q + v + D = 180 degrees   {Angle sum of triangle}   ...(2).  Adding (1) and (2) gives (p + q) + (u + v) + B + D = 360 degrees so we find A + B + C + D = 360 degrees.

Hence the angle sum of a quadrilateral is 360.


Applying Properties of Angles in Quadrilaterals

The theorems we have proved can be used to prove other theorems. They can also be used to find the values of the pronumerals in a problem.

Example 14

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

Quadrilateral ABCD has four angles of size 2x degrees, 108 degrees, 68 degrees and 130 degrees.

Solution:

2x + 108 + 68 + 130 = 360   {Angle sum of a quadrilateral}.  Solving for x we find x = 27.


Example 15

Find the value of each of the pronumerals in the kite shown here. Give reasons for your answers.

A kite has markings that show z cm = 3 cm and y cm = 7 cm.  Two angles of size 112 degrees and (3x + 4) degrees are also shown.

Solution:

3x + 4 = 112 and so x = 36     {One pair of opposite angles are equal}

Clearly, y = 7 and z = 3.


Example 16

Find the value of each of the pronumerals in the accompanying diagram. Give reasons for your answers.

A trapezium has angles of size x degrees, 130 degrees, 140 degrees and y degrees.

Solution:

x + 130 = 180     {Allied angles}.  So, x = 50.

Also, y + 140 = 180     {Allied angles}.  So, y = 40.


Example 17

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

A regular trapezium has base angles of size 4x degrees and (2x + 28) degrees.

Solution:

4x = 2x + 28     {Base angles are equal in a regular trapezium}.  Solving for x we find x = 14.


Example 18

Find the value of each of the pronumerals in the accompanying diagram. Give reasons for your answers.

A parallelogram has angles (2x + 40) degrees, 110 degrees, 70 degrees, (5y + 15) degrees.

Solution:

2x + 40 = 70   {Opposite angles of a parallelogram}.  Solving for x we find x = 15.

Also, 5y + 15 = 110   {Opposite angles of a parallelogram}.  Solving for y we find y = 19.


Example 19

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

A rhombus has sides of 17 cm and half of a diagonal of length 8 cm and half of the other diagonal of length x cm.

Solution:

The diagonals of rhombus ABCD meet at right angles at point M.

Angle AMD = 90 degrees as the diagonals of a rhombus intersect at right angles.

By Pythagoras' theorem and triangle AMD,

x squared + 8 squared = 17 squared.  Solving for x we find x = 15.


Key Terms

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


| Home Page | Order Maths Software | About the Series | Maths Software Tutorials |
| Year 7 Maths Software | Year 8 Maths Software | Year 9 Maths Software | Year 10 Maths Software |
| Homework Software | Laptop Schools | Tutor Software | Maths Software Platform | Trial Maths Software |
| Feedback | About mathsteacher.com.au | Terms and Conditions | Our Policies | Links | Contact |

Copyright 2000-2014 mathsteacher.com Pty Ltd.  All rights reserved.
Australian Business Number 53 056 217 611

Copyright instructions for educational institutions

Please read the Terms and Conditions of Use of this Website and our Privacy and Other Policies.
If you experience difficulties when using this Website, tell us through the feedback form or by phoning the contact telephone number.