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 182°.
The sum of the angles of a square is 182 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 1,440°.

Answer 1

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.

1. **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.

2. **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:
\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\]

3. **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°.**