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4.1 L) Exterior Angles: Part 1

4.1 L) Exterior Angles: Part 1

The sum of the exterior angles for any 2D shape is 360°; it does not matter how many sides the shape has.

**Triangle**

Let’s use an equilateral triangle. All of the angles in an equilateral triangle are 60°.

The exterior angles of a shape are found by continuing the lines that make up the shape.

We are able to work out the size of the exterior because we know that angles on a straight line add up to 180°. Therefore, the interior and exterior angle add up to 180°.

I am going to let the exterior angle be

*e*and we know that the interior angle is 60°.We find the value of

*e*by moving the 60 from the left side of the equation to the right. We are able to do this by doing the opposite; we take 60 from both sides.Therefore, one exterior angle of this triangle is 120°. There are 3 exterior angles, so we find the sum by multiplying 120° by 3, which gives us 360°.

**Pentagon**

I am now going to prove that the sum of the exterior angles for a pentagon is 360°. I will use a regular pentagon to do this.

Let’s first find the size of each of the interior angles, which we can do by using the formula below:

n in the formula above is the number of sides. A pentagon has 5 sides. Therefore, we sub n as 5 into the formula.

Each interior angle in a regular pentagon is 108°.

We are now able to find the size of one of the exterior angles. We are able to do this because the exterior and interior angle of a shape will add up to 180°.

We are now able to find the size of one of the exterior angles. We are able to do this because the exterior and interior angle of a shape will add up to 180°.

The interior angle of the pentagon is 108° and I am going to let the exterior angle of the pentagon be

*e*. The equation becomes:We want to find the value of

*e*, which we do by taking 108 from both sides of the equation.Each exterior angle for a regular pentagon is 72°. There are 5 exterior angles for a pentagon, so we find the sum of the exterior angles for a pentagon by multiplying 72° by 5, which is 360°, thus showing that the sum of the exterior angles is 360°.

**The Rule**

There is a rule for working out each exterior angle in a regular shape and we will be looking at the rule in the next section.