Use the angle sum of triangles to develop the interior angle sums of regular polygons formula using decomposition. Test the formula by finding the sum of the interior angles of a square.(1 point) Responses The sum of the angles of a square is 360°. The sum of the angles of a square is 360 degrees . The sum of the angles of a square is 1,080°. The sum of the angles of a square is 1,080 degrees . The sum of the angles of a square is 182°. The sum of the angles of a square is 182 degrees . The sum of the angles of a square is 1,440°.

Answer 1

To derive the formula for the sum of the interior angles of a regular polygon, we can use the principle of decomposing the polygon into triangles.

1. Understanding the Polygon: A regular polygon with \( n \) sides can be divided into \( n - 2 \) triangles. This is because we can draw \( n - 3 \) diagonals from one vertex to connect with the remaining vertices, creating \( n - 2 \) triangles.

2. Sum of Angles in Each Triangle: We know that the sum of the interior angles of a triangle is \( 180^\circ \).

3. Calculating the Total Sum for the Polygon: Therefore, for \( n - 2 \) triangles, the total sum of the interior angles of the polygon is: \[ \text{Sum} = (n - 2) \times 180^\circ \] This gives us the formula for the sum of the interior angles of a regular polygon: \[ S = (n - 2) \times 180^\circ \]

Testing the Formula: Sum of the Interior Angles of a Square

A square has \( n = 4 \) sides. We can apply the formula: \[ S = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]

Conclusion

Thus, the sum of the interior angles of a square is 360 degrees.

Response: The sum of the angles of a square is 360°.