Question

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°.