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Understanding Exterior Angles of a Polygon

CEO Khai Intela
Caption: Exterior angles of a polygon Have you ever wondered about the angles that form on the outside of a polygon? These angles, known as exterior angles, play a crucial role in understanding the properties...

Exterior angle Caption: Exterior angles of a polygon

Have you ever wondered about the angles that form on the outside of a polygon? These angles, known as exterior angles, play a crucial role in understanding the properties and characteristics of polygons. In this article, we will delve into the concept of exterior angles and explore their significance.

What are Exterior Angles?

An exterior angle is formed when we extend one side of a closed shape, such as a polygon, and consider the angle it creates with the adjacent side. For instance, consider a pentagon with five vertices. The exterior angles of this pentagon are formed by extending its adjacent sides.

Exterior angle Caption: Exterior angles of a pentagon

Some key points to note about exterior angles are:

  • They are formed on the outside or exterior of the polygon.
  • The sum of an interior angle and its corresponding exterior angle is always 180 degrees since they lie on the same straight line.
  • In the figure above, angles 1, 2, 3, 4, and 5 represent the exterior angles of the polygon.

Interestingly, the measure of exterior angles is equal in a regular polygon.

Sum of the Exterior Angles of a Polygon

If you were to travel along the vertices of a polygon in a clockwise direction, you would make turns through each exterior angle and eventually return to your starting vertex. This journey encompasses the entire perimeter of the polygon and completes one full revolution, which amounts to 360 degrees. Thus, the sum of the measures of the exterior angles of any polygon, regardless of its number of sides, is always equal to 360 degrees.

Polygon Exterior Angle Sum Theorem

The Polygon Exterior Angle Sum Theorem states that if a polygon is convex, meaning its interior angles do not exceed 180 degrees, then the sum of its exterior angles (one at each vertex) is equal to 360 degrees. Let's prove this theorem:

Proof:

Consider a polygon with n sides, also known as an n-gon. The sum of its exterior angles is denoted as N.

For any closed structure formed by sides and vertices, the sum of the exterior angles is always equal to the sum of linear pairs and the sum of interior angles. Therefore,

N = 180n - 180(n-2) N = 180n - 180n + 360 N = 360

Hence, we have proven that the sum of the exterior angles of an n-sided polygon is equal to 360 degrees.

Video Lesson on Angle Sum and Exterior Angle Property

For a more in-depth understanding of the concept of exterior angles and their properties, you can watch the following video lesson:

Exterior Angles Examples

Let's now apply our knowledge of exterior angles to solve a few examples:

Example 1: In the given figure, find the value of x.

Exterior angle example

Solution: We know that the sum of exterior angles of a polygon is 360 degrees. Thus, 70° + 60° + 65° + 40° + x = 360°. Simplifying the equation, we get x = 125°.

Example 2: Identify the type of regular polygon whose exterior angle measures 120 degrees.

Solution: Since the polygon is regular, the measure of all the interior angles is the same. Therefore, all its exterior angles measure the same as well, that is, 120 degrees. Given that the sum of exterior angles is 360 degrees and each angle measures 120 degrees, we can conclude that the polygon has 3 exterior angles. Thus, it is an equilateral triangle.

By understanding the properties of exterior angles, we can enhance our comprehension of polygons and delve deeper into their geometrical intricacies.

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