Understanding the Corollary to the Polygon Interior Angles Theorem | A Formula for Calculating the Sum of Interior Angles in Convex Polygons

Corollary to the Polygon Interior Angles Theorem

The Corollary to the Polygon Interior Angles Theorem states that the sum of the interior angles of any convex polygon with n sides is equal to (n – 2) multiplied by 180 degrees

The Corollary to the Polygon Interior Angles Theorem states that the sum of the interior angles of any convex polygon with n sides is equal to (n – 2) multiplied by 180 degrees.

To understand this corollary better, let’s break it down and explain the key concepts:

1. Convex polygon: A polygon is a closed figure with straight sides, and a convex polygon is a polygon where all of its interior angles are less than 180 degrees. A convex polygon does not have any indentations or self-intersections.

2. Interior angles of a polygon: The interior angles of a polygon are the angles formed inside the polygon by two adjacent sides. For example, if a polygon has n sides, it will also have n interior angles.

3. Polygon Interior Angles Theorem: The Polygon Interior Angles Theorem states that the sum of the interior angles of a polygon with n sides is equal to (n – 2) multiplied by 180 degrees. This theorem applies to all convex polygons.

4. Corollary: A corollary is a statement derived from a theorem, often considered a simplified or special case of the theorem. In this case, the corollary is derived from the Polygon Interior Angles Theorem.

Now, let’s see how the corollary works with an example:

Let’s consider a convex hexagon (a polygon with six sides). According to the corollary, the sum of the interior angles of this hexagon would be equal to (6 – 2) multiplied by 180 degrees, which is 4 multiplied by 180 degrees, giving us 720 degrees.

We can verify this by calculating the sum of the interior angles of the hexagon. To do this, we can divide the hexagon into triangles. A hexagon can be divided into four triangles by drawing two non-intersecting diagonals.

Each triangle has interior angles that sum up to 180 degrees. Therefore, the sum of the interior angles of the hexagon would be 4 multiplied by 180 degrees, which indeed gives us 720 degrees, confirming the corollary.

In summary, the Corollary to the Polygon Interior Angles Theorem provides a convenient formula to calculate the sum of the interior angles of any convex polygon with n sides, by using (n – 2) multiplied by 180 degrees. This corollary is derived from the Polygon Interior Angles Theorem and can be used to quickly determine the total measure of the interior angles of a convex polygon.

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