Exterior angle is the angle for the chosen side of a shape where a line is extended from the next side. In simple terms exterior angle is the angle between one side of a triangle and the extension of an adjacent side.
The sum of opposite interior angles will be equal to an exterior angle of a triangle. If an equivalent angle is taken at the vertex then the exterior angles will add to 360$^{\circ}$.

## Exterior Angles Theorem

Statement: The measure of an exterior angle of a triangle will be equal to the sum of the measures of two remote interior angles.

Proof:
The angles, $\angle$ ABC and $\angle$ BCE are the interior angles of the $\triangle$ABC.

To Prove: $\measuredangle$ BCE = $\measuredangle$ A + $\measuredangle$ B

The exterior angle $\angle$BCE is supplementary to the adjacent interior angle $\angle$ACB.

As the sum of the angles of a triangle = 180$^o$
$\measuredangle$ A + $\measuredangle$ B +$\measuredangle$ ACB = 180
$\Rightarrow$  $\measuredangle$ A + $\measuredangle$ B = 180 - $\measuredangle$ ACB   ............(1)

Again $\measuredangle$ BCE + $\measuredangle$ ACB = 180  (Linear Pair)

or $\measuredangle$ACB = 180 - $\measuredangle$BCE     ...................(2)

Substitute equation (2) in (1), we get

$\Rightarrow$  $\measuredangle$ A + $\measuredangle$ B = 180 - (180 - $\measuredangle$BCE)

= $\measuredangle$BCE

=> $\measuredangle$ BCE = $\measuredangle$ A + $\measuredangle$ B.

Hence proved.

## Exterior Angle of a Triangle

Exterior angle of a triangle is formed when any side of a triangle is extended outwards. The non straight angle outside the triangle adjacent to an interior angle will be the exterior angle of the triangle.

In the above image, the exterior angle ACD and the adjacent interior angle ACB are supplementary.
$\angle$ ACD + $\angle$  ACB = 180 $^{\circ}$

### Solved Example

Question: Find the Values of x and y.

Solution:

For the given problem we need to find x and y.
First we find 'x'.
As the sum of adjacent angles form a straight line we can write

x + 70$^{\circ}$ = 180$^{\circ}$      (Sum of adjacent angles gives a straight line)
x + 70$^{\circ}$ - 70$^{\circ}$ = 180$^{\circ}$ - 70$^{\circ}$  (Subtract 70 from both the sides)
x = 110$^{\circ}$

Now find y

The sum of angles in a triangle = 180$^{\circ}$
For $\angle$ABC we have,
y + 75$^{\circ}$ + 70$^{\circ}$ = 180$^{\circ}$
y + 145$^{\circ}$ = 180$^{\circ}$
y + 145$^{\circ}$ - 145$^{\circ}$ = 180$^{\circ}$ - 145$^{\circ}$  (Subtract 145$^{\circ}$ from both the sides)
y = 35$^{\circ}$

Therefore the values of x and y are 110$^{\circ}$ and 35$^{\circ}$ respectively.

## Exterior Angle of a Polygon

A polygon is a flat shape having straight sides. In a polygon the exterior angles add up to one full revolution only if all the angles point in the same direction around the polygon and one per vertex should be taken.

Exterior angle of a polygon is an angle which forms linear pair with one of the angles of the polygon.

In the above image we have two exterior angles at each vertex of a polygon. One exterior angle can be formed at one side of the polygon and the second at the extension of the adjacent side.

If one exterior angle is used at a vertex then the sum of exterior angles of any polygon will be 360$^{\circ}$. If there are two per vertex then the exterior angle adds up to 720$^{\circ}$.

## Exterior Angle of a Hexagon

A hexagon has six sides (even number of sides) and six vertices. Exterior angles form a linear pair with interior angle. Hexagon is made of six equilateral triangles. The exterior angle of a regular hexagon is 60$^{\circ}$ and the interior angle of a regular hexagon is 120$^{\circ}$

## Exterior Angle of a Pentagon

Pentagon has five sides and if all the angles are equal then it is regular pentagon else irregular pentagon. To find each exterior angle of a pentagon divide 360$^{\circ}$ by n (n = 5). n is the number of sides. All the exterior angles in a pentagon add up to 360$^{\circ}$.

The each interior angle of a regular pentagon is 108$^{\circ}$ and exterior angle of a regular pentagon is 72$^{\circ}$.

## Exterior Angles of a Octagon

An octagon has eight sides and as there are even number of sides the opposite sides will be parallel. Octagon can be divided into six triangles.
The exterior angle of a regular octagon is 45$^{\circ}$ and interior angle of a regular octagon is 135$^{\circ}$ as the two angles are supplementary.