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.
To develop the formula for the interior angle sum of regular polygons, we will use the angle sum of triangles and decompose the regular polygon into triangles.
Consider a regular polygon with n sides. To decompose this polygon into triangles, draw a diagonal from one vertex to all other non-adjacent vertices:
For a regular polygon with n sides, draw a diagonal from one vertex to the (n-3)th vertex. This will divide the polygon into (n-2) triangles.
Now, let's find the sum of the interior angles of each triangle:
Each triangle has three interior angles. Let's denote the sum of interior angles of each triangle as T.
Therefore, the sum of the interior angles of (n-2) triangles is (n-2)T.
However, since we are finding the sum of the interior angles of the regular polygon, we need to consider all the interior angles.
The regular polygon has n interior angles (one at each vertex). We can see that each interior angle is part of two adjacent triangles.
So, the sum of all the interior angles of the polygon is (n)A, where A represents the measure of each interior angle of the polygon.
Now, using the angle sum of triangles, we know that the sum of the interior angles of each triangle is 180 degrees.
Therefore, T = 180 degrees.
Substituting this value in our equation, (n-2)T = (n-2)(180) degrees.
Now, we equate this sum of interior angles of triangles to the sum of interior angles of the polygon: (n)A = (n-2)(180) degrees.
Simplifying this equation, we get:
nA = 180n - 360
Dividing both sides by n, we get:
A = 180 - 360/n
This is the formula for the measure of each interior angle of a regular polygon.
To test the formula, let's find the sum of the interior angles of a square.
A square is a regular polygon with 4 sides.
Using the formula, A = 180 - 360/4 = 180 - 90 = 90 degrees.
Since a square has 4 sides, the sum of the interior angles of a square is 4A = 4(90) = 360 degrees.
Therefore, the formula correctly gives us the sum of the interior angles of a square, which is 360 degrees.
In general, we can use this formula to find the sum of the interior angles of any regular polygon by substituting the appropriate value of n.