We can start with the statement that the sum of all exterior angles of a polygon will add 360 degrees.
For example, for the quadrilateral (square):
Then, each exterior angle must have a value of 360/n.
n is the number of sides.
In the case of the square, n is 4.
For a pentagon, n=5.
The interior angles are supplementary of the exterior angles, so they have a value of:
[tex]180-mExt=180-\frac{360}{n}=180\cdot(1-\frac{2}{n})[/tex]
For a quadrilateral the measure of the interior angle is 90 degrees:
[tex]180(1-\frac{2}{4})=180(1-\frac{1}{2})=180\cdot\frac{1}{2}=90[/tex]
For a pentagon (n=5), the measure of the interior angle is 108 degrees.
[tex]180(1-\frac{2}{5})=180\cdot\frac{3}{5}=108[/tex]
For a dodecagon (n=12), we have a measure of 150 degrees for the interior angle:
[tex]180(1-\frac{2}{12})=180(\frac{10}{12})=150[/tex]
