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Wize High School Geometry Textbook (Common Core) > Polygons

Exterior Angles of Polygons

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Definitions

A convex polygon is a polygon where all of the interior angles are less than 180°180\degree.

A concave polygon is a polygon where at least one of the interior angles is greater than 180°180\degree.


For any convex polygon, we can extend one side to get an exterior angle.



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What is the sum of the exterior angles of a polygon?

Recall #1:
that supplemental angles that form a straight line must add up to 180°180\degree

The interior angle and its corresponding exterior angle must add up to 180°180\degree .

So, in an n-sided polygon,
the sum of the interior angles + the sum of the exterior angles=n×180°\boxed{\text{the sum of the interior angles}~+~\text{the sum of the exterior angles}=n\times180\degree}

Recall #2:
In an n-sided polygon, the sum of the interior angles=(n−2)×180°\boxed{\text{the sum of the interior angles}=(n-2)\times180\degree}


Putting these two equations together, we see that
(n−2)×180°‘+ the sum of the exterior angles=n×180°n(180°)−2(180°) + the sum of the exterior angles=180°n180°n−360° + the sum of the exterior angles=180°n−360° + the sum of the exterior angles=0the sum of the exterior angles=360°\begin{array}{rcl} (n-2)\times 180\degree`+~\text{the sum of the exterior angles}&=&n\times180\degree\\[1em] n(180\degree)-2(180\degree)~+~\text{the sum of the exterior angles}&=&180\degree n\\[1em] 180\degree n-360\degree~+~\text{the sum of the exterior angles}&=&180\degree n\\[1em] -360\degree~+~\text{the sum of the exterior angles}&=&0\\[1em] \text{the sum of the exterior angles}&=&360\degree \end{array}


Write it Down
The sum of the exterior angles of a convex polygon is
360°360\degree

Practice: ExteriorAngles of a Polygon

Determine the measure of each missing angle below.

Practice: Exterior Angles of a Polygon

Given a regular 10-sided polygon (10-gon),

a) what is the measure of one of the exterior angles?

b) what is the measure of one of the interior angles?