To derive the formula for the sum of the interior angles of a regular polygon, we can use the concept of decomposing the polygon into triangles.
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Decomposition of the Polygon: A regular polygon with \( n \) sides can be divided into \( n - 2 \) triangles. This is done by drawing diagonals from one vertex to all other non-adjacent vertices.
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Angle Sum of Triangles: The sum of the interior angles of a single triangle is \( 180^\circ \). Therefore, if there are \( n - 2 \) triangles in the polygon, the sum of the interior angles of the polygon is: \[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \] This gives us the formula for the sum of the interior angles of a polygon with \( n \) sides: \[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
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Testing the Formula with a Square: A square has 4 sides (\( n = 4 \)). Plugging \( n = 4 \) into the formula: \[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Thus, the sum of the angles of a square is \( 360^\circ \).
Correct Response: The sum of the angles of a square is 360°.