Interior and Exterior Angles of a Triangle

First a couple of rules using this triangle:

Rule 1:The sum of the angles of a triangle (x + y + z) equals 180°

Rule 2:The exterior angle (angle w, above) of a triangle is equal to the sum of the two opposite angles (x and y, above)

Note that an exterior angle lies **outside** the triangle and is made by extending one sides of the triangle.

## This video shows these 2 rules in action:

Constructing Triangles

### SAS and ASA

SAS: If you know the length of 2 sides and one angle you can construct a full triangle.

ASA:If you know 2 angles and the length of one side between those angles you can complete a triangle.

Here’s a video showing the construction of a triangle based on SAS:

Here’s a video showing the construction of a triangle using ASA:

## Quadrilaterals

A quadrilateral is a shape formed by 4 straight lines. These are all types of quadrilateral:

The above were some special, named quadrilaterals. All quadrilaterals have certain attributes:

- Quadrilaterals are 2D shapes bounded by four straight lines.
- The diagonal – a straight line joining two opposite vertices or corners– divides the quadrilateral into two triangles.

We know that the sum of a triangle’s interior angles adds up to 180° so it follows that if quadrilaterals can be divided into 2 triangles then the sum of a quadrilateral’s interiors angles add up to 180° plus 180° = 360°

## Polygons

There are a number of rules about ploygons. Quite often questions will ask you to use these rules to calculate interior or exterior angles.

A polygon is a two dimensional shape bounded by straight lines, so a triangle is a polygon.

Regular polygons have sides that are the same length and all the angles in a regular polygon are the same size.

These are all regular polygons:

3 sides — Equilateral triangle

4 sides — Square

5 sides — Regular Pentagon

6 sides — Regular Hexagon

7 sides — Regular Heptagon

8 sides — Regular Octagon

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

In a regular polygon all the exterior angles are the same size and can be calculated using this rule:

Exterior angle of a regular polygon = 360°/number of sides

So for a regular pentagon (5 sides) each exterior angle will be:-

360°/5 = 72°

Quadrilaterals (see above) are four-sided polygons. We saw that a quadrilateral could be divided into 2 triangles by drawing a line from one vertex (corner) to the opposite vertex and that this proved that the interior angles of a quadrilateral were 360° (180° + 180°).

In a similar way any polygon can be divided into a number of triangles. A quadrilateral (4 sides) can be divided into 2 triangles, a pentagon (5 sides) can be divided into 3 triangles and hexagon can be divided into 4 triangles. Can you see the pattern or rules that’s emerging? –

The sum of the interior angles of a polygon is (number of sides — 2) x 180°

When you consider a polygon’s vortex and it’s interior and exterior angle, note that the interior and exterior angles lie on a straight line so:

Interior angle + Exterior angle = 180°

**OR**

Interior angle = 180° — Exterior angle

**OR**

Exterior angle = 180° — Interior angle.