- Identify the exterior angles of convex polygons.
- Find the sums of exterior angles in convex polygons.
This lesson focuses on the exterior angles in a polygon. There is a surprising feature of the sum of the exterior angles in a polygon that will help you solve problems about regular polygons.
Exterior Angles in Convex Polygons
Recall that interior means inside and that exterior means outside. So, an exterior angle is an angle on the outside of a polygon. An exterior angle is formed by extending a side of the polygon:
There are two possible exterior angles for any given vertex on a polygon. In the figure above, one set of exterior angles is shown, the set in the counter-clockwise direction. The other set of exterior angles would be formed by extending each side of the polygon in the opposite (clockwise) direction. However, it does not matter which exterior angles you use because their measurement will be the same on each vertex. Look closely at one vertex below, where we draw both of the possible exterior angles:
In the above diagrams, both exterior angles are drawn separately. On the next page, both exterior angles on a single vertex are drawn together:
As you can see, the two exterior angles at the same vertex are vertical angles.
Since vertical angles are congruent, the two exterior angles possible around a single vertex are congruent.
The clockwise exterior angle and the counter-clockwise exterior angle at the same vertex are _______________________________ .
Additionally, because the exterior angle will be a linear pair with its adjacent interior angle, it will always be supplementary to that interior angle.
1. True or False: Vertical angles are supplementary.
2. True or False:
Exterior angles are on the outside of a polygon and are formed when you extend one side of the polygon.
3. Name the 2 sets of exterior angles:
5. Complete the sentence:
Since the interior angle and the exterior angle at the same vertex of a polygon form a linear pair, ...
Summing Exterior Angles in Convex Polygons
By now you might expect that if you add up various angles in polygons, there will be some sort of pattern or rule.
There is also a rule for exterior angles in a polygon.
Let’s begin by looking at a triangle:
To find the exterior angles at each vertex, extend the segments and find angles supplementary to the interior angles:
The sum of these three exterior angles is:
Let’s see what happens with another shape.
To compare, examine the exterior angles of a rectangle:
Find the sum of the 4 exterior angles in a rectangle:
Exterior Angle Sum
On the other hand, we already saw that the sum of the interior angles was:
Putting these together we have:
1. True or False:
In a polygon, an interior angle and one of the exterior angles at the same vertex are supplementary.
3. Fill in the blanks:
4. What is the rule for the sum of exterior angles in a polygon? Describe.
4 of the 5 exterior angles on this polygon have their measurements labeled: