To prove the Polygon Interior-Angle Sum Theorem, we start with the definition of a polygon and its properties. A polygon is a closed figure formed by a finite number of straight line segments connected end-to-end. The theorem states that the sum of the interior angles of a polygon with \( n \) sides is equal to \( (n - 2) \times 180 \) degrees.
Proving the Interior-Angle Sum Theorem
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Base Case: Let's consider a simple polygon with three sides (a triangle). A triangle has three angles, and the sum of the angles in a triangle is: \[ (3 - 2) \times 180 = 1 \times 180 = 180 \text{ degrees} \] Thus, the theorem holds for \( n = 3 \).
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Inductive Step: Assume that the theorem holds for all polygons with \( k \) sides, where \( k \geq 3 \). Thus, we assume: \[ \text{The sum of the interior angles of a polygon with } k \text{ sides is } (k - 2) \times 180 \text{ degrees.} \]
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Adding a New Side: Now, consider a polygon with \( k + 1 \) sides. We can form this polygon by taking a \( k \)-sided polygon and adding one more side. To do this, we can draw one diagonal from one of the vertices to another non-adjacent vertex. This diagonal divides the \( k \)-sided polygon into two triangles.
- A \( k \)-sided polygon has a sum of interior angles of \( (k - 2) \times 180 \) degrees according to our assumption.
- Each triangle has a sum of interior angles equal to \( 180 \) degrees.
- By adding one new triangle (created by the new side and the diagonal), we add \( 180 \) degrees to the sum of the angles.
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Calculate the New Sum: Therefore, the total sum of the angles in the \( (k + 1) \)-sided polygon is: \[ (k - 2) \times 180 + 180 = (k - 2 + 1) \times 180 = (k - 1) \times 180 \]
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Conclusion: This establishes that the sum of the interior angles of a polygon with \( k + 1 \) sides is indeed: \[ (k + 1 - 2) \times 180 = (k - 1) \times 180 \text{ degrees.} \] Hence, by the principle of mathematical induction, the sum of the interior angles of a polygon with \( n \) sides is: \[ (n - 2) \times 180 \text{ degrees.} \]
Answer to the Prompt
- Thus, the angle sum of all the triangles is the product: \[ \text{Sum} = (n - 2) \times 180 \text{ degrees.} \]