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
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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.
To develop the formula for the interior angle sum of regular polygons, we can start by using the angle sum of triangles.
1. Begin with a regular polygon.
2. Draw lines from one vertex to all the other vertices, creating triangles.
3. Count the number of triangles formed.
Since there are n vertices in a regular polygon, there will also be n triangles formed. Each triangle has an interior angle sum of 180 degrees. Therefore, the total sum of the interior angles of the regular polygon can be found by multiplying the number of triangles (n) by the interior angle sum of each triangle (180 degrees):
Interior angle sum of regular polygon = n * 180 degrees.
To test this formula with a square, we know that a square has 4 sides and vertices. Therefore, the interior angle sum of a square can be calculated as:
Interior angle sum of a square = 4 * 180 degrees
= 720 degrees.
Thus, the sum of the interior angles of a square is 720 degrees.
To develop the interior angle sum formula for regular polygons using decomposition, we can start with a triangle and gradually increase the number of sides.
1. Start with a triangle: A triangle has three angles, and the sum of its interior angles is always 180 degrees.
2. Extend to a quadrilateral: A quadrilateral is essentially two triangles put together. So, the sum of its interior angles is twice that of a triangle, which is 2 * 180 = 360 degrees.
3. Expand to a pentagon: A pentagon has five sides, and we can decompose it into three triangles. So, the sum of its interior angles is three times that of a triangle, which is 3 * 180 = 540 degrees.
4. Continue to a hexagon: A hexagon has six sides, and we can decompose it into four triangles. So, the sum of its interior angles is four times that of a triangle, which is 4 * 180 = 720 degrees.
5. Generalize for a regular polygon with n sides: From the above patterns, we can observe that for any regular polygon with n sides, we can decompose it into (n-2) triangles. So, the formula for the sum of its interior angles is (n - 2) * 180 degrees.
Now, let's test this formula by finding the sum of the interior angles of a square.
For a square, n = 4. Substituting this value into the formula, we get (4 - 2) * 180 = 2 * 180 = 360 degrees.
Therefore, the sum of the interior angles of a square is 360 degrees.
Note: The answer provided may vary based on the unit used for measuring angles (degrees or radians). In this case, we are assuming degrees.