The Polygon Exterior Angle Sum Theorem: Explained and Proven

Polygon Exterior Angle Sum Theorem

The Polygon Exterior Angle Sum Theorem states that the sum of the measures of the exterior angles of any polygon is always 360 degrees

The Polygon Exterior Angle Sum Theorem states that the sum of the measures of the exterior angles of any polygon is always 360 degrees.

To understand this theorem, let’s start by defining what an exterior angle of a polygon is. An exterior angle is formed by extending one of the sides of the polygon outwards. For example, in a triangle, if you extend one of the sides, the angle formed by the extension and the adjacent side is the exterior angle.

Now, let’s visualize this theorem with an example of a polygon. Let’s consider a polygon with n sides. To find the sum of the exterior angles, we can start by examining the sum of the angles at each vertex.

In any polygon, the sum of the interior angles at each vertex is always 180 degrees. This means that if you add up the measures of all the interior angles of a polygon, the sum will be (n – 2) * 180 degrees, where n represents the number of sides of the polygon.

Now, each interior angle and its adjacent exterior angle form a straight line, which measures 180 degrees. In other words, the interior angle and its adjacent exterior angle are supplementary. Therefore, the sum of each pair of interior and exterior angles at each vertex is always 180 degrees.

Since there are n vertices in the polygon, the sum of all the pairs of interior and exterior angles would be n * 180 degrees.

However, remember that the sum of the exterior angles is the same as the sum of these pairs of interior and exterior angles. Therefore, we can conclude that the sum of the measures of the exterior angles of any polygon is always n * 180 degrees.

But, we also know that the sum of the measures of the exterior angles is equal to 360 degrees (as stated by the Polygon Exterior Angle Sum Theorem). So, we can equate these two expressions:

n * 180 degrees = 360 degrees.

We can solve this equation by dividing both sides by 180:

n = 360 degrees / 180 degrees = 2.

This tells us that the sum of the measures of the exterior angles of any polygon is always 2 times the number of sides. In other words, the number of sides of the polygon is half the sum of the measures of the exterior angles.

For example, in a triangle (3 sides), the sum of the exterior angles is 3 * 180 degrees = 540 degrees, which satisfies the theorem. In a pentagon (5 sides), the sum of the exterior angles is 5 * 180 degrees = 900 degrees, which also satisfies the theorem.

So, no matter the number of sides in a polygon, the sum of the measures of its exterior angles will always equal 360 degrees. This is a fundamental property of polygons known as the Polygon Exterior Angle Sum Theorem.

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