Page 45 - Cambridge+Checkpoint+Mathematics+Coursebook+9

5.2 More polygons

5.2 More polygons

What do you know about the angles of a pentagon that is not regular? Exterior
!e external angles still add up to 360°, as long as all the interior angles angles

are less than 180°. !e explanation is the same as for a regular pentagon.
What about the internal angles?
You can divide any pentagon into three triangles.

!e angles of the three triangles make up the "ve Interior
angles
angles of the pentagon.

!e sum of the angles of the pentagon is 3 × 180° = 540°.
You can use this method for any polygon. This is a general result.
Polygon, N sides Sum of exterior angles = 360° Sum of interior angles = (N − 2) × 180°

Worked example 5.2

One angle of a hexagon is 90°. The other angles are all equal. How big are the other angles?

A hexagon has six angles that add up to 720°. N = 6; (N − 2) × 180 = 4 × 180 = 720
720 − 90 = 630
The other angles are each 630 ÷ 5 = 126°.

) Exercise 5.2

1 Work out the sum of the interior angles of:
a a heptagon (7 sides) b a nonagon (9 sides) c a decagon (10 sides).

2 Five of the interior angles of a hexagon are 90°, 100°, 110°, 120° and 130°.
a Work out the other interior angle of the hexagon.
b Calculate the external angles of the hexagon and show that they have the correct total.

3 Four of the interior angles of a hexagon are 128°. The other two angles are equal.
How large are the other two angles?

4 Xavier has a rectangular piece of card. He cuts off the four corners.
What do the angles of the remaining shape add up to?
5 Read what Alicia says.

The angles of this pentagon are
100°, 105°, 72°, 126° and 127°.
Explain why she must be wrong.

6 The interior angles of a polygon add up to 1800°.
How many sides does it have? Give a reason for your answer.

7 a Four of the interior angles of a pentagon are 105°. Work out the fifth angle.
b Can a pentagon have four right angles? Give a reason for your answer.

44 5 Shapes