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9. Why is the sum of the interior angles of a polygon always a multiple of 180 ?

Ask by Warner Maxwell. in the United States
Feb 27,2025

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Answer

The sum of the interior angles of any polygon is always a multiple of 180° because dividing the polygon into triangles results in a total angle sum of (n – 2) × 180°, where n is the number of sides.

Solution

Consider any n-sided polygon. One common way to understand why the sum of its interior angles is always a multiple of 180° is to divide (or "triangulate") the polygon into triangles. Here's how it works: • Pick one vertex of the polygon. • Draw diagonals from that vertex to every other non-adjacent vertex. • This splits the polygon into (n – 2) triangles. • Since each triangle has an interior angle sum of 180°, the total sum for all these triangles is (n – 2) × 180°. Because (n – 2) × 180° is clearly a multiple of 180, the sum of the interior angles of any polygon is always a multiple of 180°.

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Every polygon can be divided into triangles, and the sum of the interior angles of a triangle is always 180 degrees. For example, a quadrilateral can be divided into two triangles, creating two sets of 180 degrees, totaling 360 degrees, and so forth for polygons with more sides. Thus, you can generalize this to any polygon by using the formula (n-2) × 180, where n is the number of sides, proving the sum is always a multiple of 180. This concept has real-world applications in fields like architecture and engineering. When designing structures, understanding the properties of polygons, including the sum of their angles, helps professionals create stable and aesthetically pleasing structures. Whether it’s the roof of a building or the layout of a park, the knowledge of angle sums plays a critical role in ensuring the designs are not only functional but also visually appealing!

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