To prove the Polygon Interior-Angle Sum Theorem, which states that the sum of the measures of the interior angles of a polygon with \( n \) sides is \( (n - 2) \times 180^\circ \), we can use a method involving triangulation of the polygon.
Proof by Triangulation:
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Understand the Structure of Polygons:
- A polygon with \( n \) sides is called an \( n \)-gon.
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Dividing the Polygon into Triangles:
- We can divide a polygon with \( n \) sides into \( n - 2 \) triangles by drawing diagonals from one vertex to all other non-adjacent vertices.
- For example, from one vertex of the polygon, you can connect to \( n - 3 \) other vertices (since one vertex is the starting point and two are adjacent and can't be connected directly).
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Calculating the Angles in the Triangles:
- Each triangle has a sum of interior angles equal to \( 180^\circ \).
- Since we have \( n - 2 \) triangles formed, the sum of the measures of the angles in these triangles is: \[ (n - 2) \times 180^\circ \]
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Relating Triangle Angles to Polygon Angles:
- The angles of the triangles formed by triangulating the polygon are exactly the interior angles of the polygon.
- Thus, the total sum of the interior angles of the polygon is equal to the total sum of the angles from the triangles.
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Conclusion:
- Therefore, we conclude that the sum of the measures of the interior angles of a polygon with \( n \) sides is: \[ (n - 2) \times 180^\circ \]
Final Statement:
Thus, the sum of the measures of the interior angles of a polygon with \( n \) sides is \( (n - 2) \times 180^\circ \).