Cambridge International Mathematics

Lines, angles and polygons (Chapter 4) 103

5 Because of its symmetry, a regular pentagon can be constructed

from five isosceles triangles.

a Find the size of angleμat the centre O.

b Hence, findÁ.

c Hence, find the measure of one interior angle such as ABC.b

6 Repeat question 5 but use a regular decagon. Remember that a

decagon has 10 sides.

We have already seen that the sum of the interior angles of a triangle is 180 o.

Now consider finding the sum of the angles of a quadrilateral.

We can construct a diagonal of the quadrilateral as shown.

The red angles must add to 180 oand so must the green angles.

But these 6 angles form the 4 angles of the quadrilateral.

So, the sum of the interior angles of a quadrilateral is 360 o.

We can generalise this process to find the sum of the interior angles of any polygon.

Discovery 1 Angles of ann-sided polygon#endboxedheading

What to do:

1 Draw any pentagon ( 5 -sided polygon) and label one of its

vertices A. Draw in all the diagonals from A.

2 Repeat 1 for a hexagon, a heptagon ( 7 -gon), an octagon,

and so on, drawing diagonals from one vertex only.

3 Copy and complete the following table:

Polygon

Number

of sides

Number of

diagonals fromA

Number of

triangles

Angle sum

of polygon

quadrilateral 4 1 2 2 £ 180 o= 360o

pentagon

hexagon

octagon

20 -gon

4 Copy and complete the following statement:

“The sum of the interior angles of anyn-sided polygon is ......£ 180 o.”

D THE INTERIOR ANGLES OF A POLYGON [4.4]

A

B

C

f°

q°

A

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Y:\HAESE\IGCSE01\IG01_04\103IGCSE01_04.CDR Monday, 15 September 2008 11:55:57 AM PETER