Corollary to the Polygon Angle-Sum Theorem
The corollary to the Polygon Angle-Sum Theorem states that the sum of the interior angles in a polygon with n sides is equal to (n-2) times 180 degrees
The corollary to the Polygon Angle-Sum Theorem states that the sum of the interior angles in a polygon with n sides is equal to (n-2) times 180 degrees. This corollary is derived from the Polygon Angle-Sum Theorem, which states that the sum of the interior angles in any polygon is equal to (n-2) times 180 degrees, where n represents the number of sides.
To understand this corollary, let’s consider a polygon with n sides. To find the sum of the interior angles, we can draw (n-2) diagonals from any single vertex. This will divide the polygon into (n-2) triangles.
Since the sum of the interior angles in each triangle is 180 degrees (by the Triangle Angle Sum Theorem), we can conclude that the sum of the interior angles in the polygon will be (n-2) times 180 degrees, as there are (n-2) triangles.
To illustrate this, let’s take a triangle as an example. A triangle has 3 sides and (3-2) = 1 triangle. The sum of the interior angles in a triangle is (1 * 180) = 180 degrees, which matches the corollary.
Similarly, if we consider a quadrilateral with 4 sides, we can draw (4-2) = 2 diagonals, which will divide the polygon into 2 triangles. The sum of the interior angles in each triangle is 180 degrees, so the sum of the interior angles in the quadrilateral is (2 * 180) = 360 degrees, which aligns with the corollary.
You can apply the same reasoning to any polygon, and you will find that the sum of the interior angles is always equal to (n-2) times 180 degrees.
In summary, the corollary to the Polygon Angle-Sum Theorem states that the sum of the interior angles in a polygon with n sides is (n-2) times 180 degrees. This corollary is derived from dividing the polygon into (n-2) triangles and applying the Triangle Angle Sum Theorem to each triangle.
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