A quadrilateral is a four-sided polygon. Quadrilaterals can be different shapes but they all have four vertices (the point where two sides meet to form an angle) and two diagonals. The sum of the interior angles in a quadrilateral is always 360º.

Quadrilaterals fall into different groups depending on their parallel sides.

Parallelograms are quadrilaterals with both pairs of opposite sides parallel to each other. Furthermore, all parallelograms have the following properties:

• Opposite sides are the same length.

• Opposite angles are equal.

• The diagonals cross each other at their mid-point.

A trapezium is a quadrilateral  with only one pair of parallel sides.

A trapezoid is a quadrilateral with no parallel sides.

Drag the points B, C and D with the mouse to make three of each type of quadrilateral: three parallelograms, three trapeziums and three trapezoids.

In each case write down the length of the sides and measurements of the angles in your notebook.

Check that the three properties listed above are true in each parallelogram.

As we have already said, parallelograms are quadrilaterals with two pairs of opposite sides. Parallelograms can be divided into three further groups:

• Rectangles, which have four equal angles (right angles).

• Rhombuses, which have four equal sides.

• Squares, which have four equal angles (right angles) and four equal sides.

• Strictly speaking, all other paralellograms which are not rectangles, rhombuses or squares are called rhomboids.

In the following window move the three points B, C and D of the quadrilateral to form the paralellograms described below.

In each case, write down the length of the diagonals and the measurements of the angles formed between them.

Measure the length of the diagonals in the rectangles and squares.

In the rhombuses work out the relationship between half of the length of the diagonal and the length of its sides.

a) A rectangle with sides of 5 and 8 units long. What are the sizes of the four angles of the rectangle? (Use Pythagoras' theorem to calculate the length of the diagonals).

b) A rectangle with sides of 10 and 7 units long.

c)  A rhombus with sides of 7 units long and one of the angles 30º. Write down the sizes of the other angles in the rhombus. Note that in a rhombus the diagonals are always perpendicular and divide the rhombus up into four equal right-angled triangles. Apply Pythagoras' theorem to check the measurements of these triangles.  

d) A rhombus with sides of 10 units long and one of the angles 25º.Write down the sizes of the other angles in the rhombus. Apply Pythagoras' theorem to check the measurements of the triangle formed by one of the sides and the diagonals of the rhombus.  

e) A square with sides of 8 units long. Calculate the length of the diagonals.

f) A square with sides of 12 units long. Calculate the length of the diagonals.

g) A rhomboid with adjacent sides of 9 and 7 units long and one of the angles 30º. Write down the sizes of the other angles in the rhomboid.

h) A rhomboid with adjacent sides of 6 and 10 units long and one of the angles 40º. Write down the sizes of the other angles in the rhomboid.

i) What is the connection between the lengths of the diagonals in different kinds of paralellograms?

j) Can you see anything characteristic about the angle formed by the diagonals in each kind of parallelogram? 

In a rectangle we know that one side is 4 cm long and its diagonals are 5cm long. How long is the other side? 

Once you have solved the problem in your exercise book form a rectangle in the window above with these measurements to check your answer.

In the following window form the trapeziums described below. In each case write down  the size of the angles in your notebook.

a) A right-angled trapezium whose parallel sides are 8 and 13 long and the side adjacent to the right angle is 6 units long.  

b) A right-angled trapezium whose parallel sides are 4 and 10 units long and the side adjacent to the right angle is 5 units long.

c) An isosceles trapezium whose parallel sides are 8 and 13 units long and whose non-parallel sides are 6 units long.

d) An isosceles trapezium whose parallel sides are 4 and 10 units long and whose non-parallel sides are 5 units long.

e) A scalene trapezium whose parallel sides are 8 and 13 units long and whose non-parallel sides are 5 and 7 units long.

A kite is a special kind of trapezoid: its diagonals are perpendicular and cross at the mid-point of one of them. 

Move points A, B and C of the kite in the following window to form three different kites: the first with diagonals of 6 and 8 units long, the second of 10 and 5 units long and the third of 12 and 7 units long.

Write down the following information about each kite in your notebook: the length of the sides and of the diagonals and the size of the angles.

Work out the connection between the measurements of the sides and the angles of a kite and write it down in your notebook.

The diagonals of a rhombus are 12 and 8 cm long. Work out the length of its sides. 

Once you have solved the problem in your notebook form a rhombus with these measurements in the window above to check your result.