## IGCSE Maths Lesson - Polygons and Triangles

### Polygons and Triangles

#### Polygons

A polygon is a closed shape made by many straight lines. "Poly" meaning many.

#### Triangles

The polygon with the least number of sides is a triangle. A triangle is a shape with three sides, each of which is a straight line.

There are three angles in a triangle and they add together to make 180°. These angles are called the interior angles.

### Types of Triangle

#### Categorised by angle:

**Acute Angled**– this triangle has three acute angles.**Obtuse Angled**- this triangle has one obtuse angle (and two acute angles).**Right Angled**– this triangle has a right angle and two acute angles. The side opposite the right angle, which is the longest side, is called the hypotenuse.

#### Categorised by sides:

**Scalene**– this triangle has sides all of different lengths, the angles are also all different.**Isosceles**– this triangle has two sides that are equal in length. Two of its angles are also equal. (Greek meaning equal legs)**Equilateral**– this triangle has all three sides of equal length. “Equi” – meaning equal “lateral” – side

All of its angles are also equal. As the angles of a triangle all add up to 180°, each angle in an equilateral triangle is 180 ÷ 3 = 60°.

If any of the sides of a triangle are equal, this can be shown by a small line through each of the equal sides. For example, the following triangle is an isosceles triangle.

### Exterior Angles of triangles

An exterior angle of a triangle is one that is formed when any of the sides is extended. For example,

The exterior angle is equal to the sum of the two opposite angles, i.e. d = a + b.

You will need to understand the proof that this is so. We can prove it by using some of the facts that we have already learnt.

The angles of a triangle add up to 180°, and the angles on a straight line add up to 180°

Therefore, a + b + c =180° and c + d =180°

Therefore, a + b + c = c + d

Subtract c from both sides: a + b = d

### Interior Angles of a triangle

We have already learnt that the interior angles of a triangle add up to make 180°. You will need to understand (but not reproduce) the proof of this fact:

Take the triangle ABC with angles x, y and z

Draw a line parallel to AB and call it CD, and extend BC to a new point E

Angle ACD and x are alternate angles, so ACD = x

Angle DCE and y are corresponding angles, so DCE = y

BCE is a straight line, therefore z + ACD + DCE = 180

Substituting x and y in gives x + y + z = 180°, i.e. the angles in a triangle add up to 180°

### Quadrilaterals

A quadrilateral is a polygon with four sides. “Quad” meaning four.

There are four angles in a quadrilateral and they add together to make 360°.

### Types of Quadrilateral

#### Square

Properties of a square:

- four equal sides
- four right angles
- opposite sides are parallel to each other
- adjacent sides are perpendicular to each other
- diagonals of equal length which bisect each other at 90° (bisect means cut in half)

#### Rectangle

Properties of a rectangle:

- opposite sides are equal
- four right angles
- opposite sides are parallel to each other
- adjacent sides are perpendicular to each other
- diagonals of equal length and bisect each other

#### Parallelogram

Properties of a parallelogram:

- opposite sides are equal
- opposite angles are equal
- opposite sides are parallel to each other
- The diagonals will bisect each other

#### Rhombus

Properties of a rhombus:

- Four equal sides
- opposite angles are equal
- opposite sides are parallel to each other
- The diagonals will bisect each other at 90°
- Sometimes referred to as a diamond

#### Trapezium

Properties of a trapezium:

- One pair of parallel sides

#### Kite

Properties of a Kite:

- Two pairs of equal adjacent sides
- One pair of equal opposite angles
- The diagonals will bisect each other at 90°

You may be asked to use the properties of these quadrilaterals to solve problems.

#### Example 1

Work out the size of angle b

We can see that this is a parallelogram because the opposite sides are parallel. We know that in a parallelogram opposite angles are equal, therefore angle a = 75°

and the angle opposite to b is equal to b.

We also know that the four angles of a quadrilateral add up to make 360°

So, 75 + 75 + 2b = 360

2b = 360 – 150, 2b = 210, b = 105°

#### Example 2

Work out the size of angles a and b

Extend line DC to create the external angle

Then z = 180° - 75°

z = 105°

Now we need to remember one of the facts about angles and parallel lines. AC is a transversal to the parallel lines AB and CD.

Using alternate angles, z = x. So x = 105°.

Now we can use the fact that the four angles of a quadrilateral add up to make 360°

75 + 110 +105 + y = 360°; y = 70°

If you cannot solve a problem by looking only at the interior angles, try extending one of the lines to form an exterior angle that you can work out and then use the facts that you know about angles and parallel lines.

### Polygons with more than four sides

#### Examples

- 7 sides – heptagon
- 8 sides – octagon
- 9 sides – nonagon
- 10 sides - decagon

### Interior and Exterior angles of a polygon

Angles inside a polygon are called interior angles. When the side of a polygon is extended the angle now formed is called an exterior angle.

The point where the sides of a shape meet is called the vertex. The plural of vertex is vertices.

At each vertex of a polygon, Interior angle + exterior angle = 180°

### Sum of the interior angles of a polygon

Polygons can be divided up into triangles by the diagonals from 1 vertex.

For example, a rectangle can be divided up into two triangles as shown

The sum of interior angles in a triangle is 180°, therefore the sum of the interior angles in a rectangle is 2 × 180 = 360°

A pentagon can be split into 3 triangles by drawing diagonals from one of the vertices.

The interior angles of a pentagon therefore = 3 × 180 = 540°

In general, for any polygon of n sides, the sum of the interior angles is (n - 2) × 180°

### Sum of the exterior angles of a polygon

The sum of the exterior angles for any polygon is 360°

a + b + c + d +e = 360°

### Regular Polygons

If a polygon has all sides and all angles equal it is said to be regular.

Sometimes you will be asked to find the number of sides in a regular polygon given an interior or exterior angle. We can rearrange this formula to help us.

For example,

A regular polygon has interior angles of 120°. How many sides does it have?

Interior angle + exterior angle = 180°

Exterior angle = 180 – 120 = 60°

#### How to construct your own triangles using a protractor, compass and ruler

If you have ever had to plan/visualise your journey on a map or chart you may have used a similar technique to the one you’ll see below, if not this may be the 1st time you’ve seen this, I’m terrible at drawing so I love this simple method!