Question

For a regular n-gon:
a. What is the sum of the measures of its angles?
b. What is the measure of each angle?
c. What is the sum of the measures of its exterior angles, one at each vertex?
d. What is the measure of each exterior angle?
e. Find the sum of your answers to parts b and d. Explain why this sum makes sense.

Answer 1

is the geometry?
ok a nine sided polygon?

Answer 2

a) If you join each vertex of the n-gon to the center it will make n triangles.
The sum of the angles of a traingle is180 degrees.
N triangles will have 180*n degrees.
At the center the angle made by all the lines joining the vertices to the center is 360 degrees.
So the interior angles of a n-gon is: 180*n - 360 = 180(n - 2) degrees.

Answer 3

when it is regular
sum of angles divided by n
so (n-2)*180
interior angle =---------
n

when you want extrior angle of regular n-gon
it is sufficient to subtract 180 - interior
so ext angle =180 -(n-2)*180/n =360/n

Answer 4

a) Sum of the measures of its angles = 180(n-2) (see my first reply)
b) Measure of each angle = 180(n-2) / n
c) The exterior angles of all polygons add up to 360 degrees.
d) Measure of each exterior angle = 360 / n
e) add answers to b and d: 180(n-2)/n + 360/n =
{ 180n - 360 + 360 } / n = 180.
This makes sense because the exterior angle and the interior angle at each vertex forms a linear pair and therefore the sum of the angles will be 180 degrees.

Answer 5

Thanks :)

Answer 6

yw.