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)

The sum of the angles of a square is 360 deg

O The sum of the angles of a square is 182 ^ * .

The sum of the angles of a square is 1, 440 deg

The sum of the angles of a square is 1 ,080^ .

3 answers

To develop the interior angle sums of regular polygons formula using decomposition, we start with the concept that the sum of the angles in any triangle is 180 degrees. Then, we can decompose a regular polygon into triangles by drawing diagonals from one vertex to all the other vertices, except for the adjacent vertices. This decomposes the polygon into n - 2 triangles, where n is the number of sides of the polygon.

For example, in a square, we can draw two diagonals that intersect at the center, forming four congruent triangles. Therefore, the square can be decomposed into four triangles.

Since each triangle has an interior angle sum of 180 degrees, and the square is decomposed into four triangles, the sum of the interior angles of a square is 4 * 180 degrees, which is equal to 720 degrees.

Therefore, the formula for the sum of the interior angles of a regular polygon with n sides is:

Sum_of_Interior_Angles = (n - 2) * 180 degrees

Now, let's test the formula by finding the sum of the interior angles of a square (n = 4):

Sum_of_Interior_Angles = (4 - 2) * 180 degrees = 2 * 180 degrees = 360 degrees

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

So what the answer

The answer is 360 degrees.