One exterior angle of a regular polygon measures 36. What is the sum of the polygon's interior angle measures?

![https://useruploads.socratic.org/oEjsbceqTS2oheH5wQMH_polygon_exterior_angle.jpg)

In the diagram above, let your exterior angle be #36#. Now your interior angle and exterior angle with the line extended add to equal to 180 degrees since the line drawn is a straight line. Therefore, if one exterior angle of the regular polygon is 36, then the interior angle is #180-36=144#

The angle sum of any polygon is given by #180(n-2)# where n is the number of sides of the polygon

Therefore, the angle sum of the polygon is equal to the one interior angle multiplied by the number of angles there are. Now since the number of angles is equivalent to the number of sides, we can write it like this:
#180(n-2)=144n#
#180n-360=144n#
#36n=360#
#n=10#

Since the polygon has 10 sides, then it must be a decagon.

Now, putting #n=10# back into the equation #180(n-2)# gives:

#180(10-2)=180times8=1440#

Let the number of sides of regular polygon be #n#

Exterior angle of regular polygon is #E= 360/n ; E=36^0#

#:. n=360/E=360/36=10#. So it is regular Decagon having #10#

equal sides. Sum of polygon's interior angles is

#sum i= (n-2)* 180 = (10-2)*180= 1440^0# [Ans]