Question 118697
1) You can use the following facts about polygons to help solve this problem:
a) The sum of an interior angle and an exterior angle is 180 degrees.{{{A[i]+A[e] = 180}}}
b) The measure of an interior angle of a regular polygon of n sides is given by:{{{A[i] = (n-2)180/n}}} This works only for "regular" polygons.
In this problem, you have:
{{{A[i] = A[e]+100}}} "The interior angle is 100 degrees more than the exterior angle"
So you can write:
{{{A[i]+A[e] = 180}}} Substitute {{{A[i] = A[e]+100}}}
{{{(A[e]+100)+A[e] = 180}}} Simplify.
{{{2A[e]+100 = 180}}} Subtract 100 from both sides.
{{{2A[e] = 80}}} Divide both sides by 2.
{{{A[e] = 40}}} The measure of an exterior angle is 40 degrees.
{{{A[i] = 180-A[e]}}}
{{{A[i] = 180-40}}}
{{{A[i] = 140}}} The measure of an interior angle is 140 degrees.
To find the number of sides (n) in this regular polygon, use:
{{{A[i] = (n-2)180/n}}} Substitute {{{A[i] = 140}}} to get:
{{{140 = (n-2)180/n}}} Simplify and solve for n, the number of sides.
{{{140 = (180n-360)/n}}} Multiply both sides by n.
{{{140n = 180n-360}}} Add 360 to both sides.
{{{140n+360 = 180n}}} Subtract 140n from both sides.
{{{360 = 40n}}} Divide both sides by 40.
{{{9 = n}}}
The regular polygon has 9 sides and this is called a "Nonagon"

2) In this problem, you have: "The ratio of an interior angle to an exterior angle is 5:1 Or the interior angle is five times the exterior angle.
Starting with: The sum of the interior and exterior angles is 180 degrees.
{{{A[i]+A[e] = 180}}} Substitute: {{{A[i] = 5*A[e]}}}
{{{5*A[e]+A[e] = 180}}} Simplify and solve for {{{A[e]}}}
{{{6*A[e] = 180}}} Divide both sides by 6.
{{{A[e] = 30}}}
The exterior angle is 30 degrees.
{{{A[i] = 180-A[e]}}} Substitute {{{A[e] = 30}}}
{{{A[i] = 180-30}}}
{{{A[i] = 150}}}
The interior angle is 150 degrees.
Check:
{{{A[i]/A[e] = 5/1}}} Substitute{{{A[i] = 150}}} and {{{A[e] = 30}}}
{{{150/30 = 5/1}}} Reduce the fraction on the left side.
{{{5/1 = 5/1}}}
To find the number of sides (n), use:
{{{A[i] = (n-2)180/n}}} Substitute {{{A[i] = 150}}} to get:
{{{150 = (n-2)180/n}}} Simplify and solve for n. Multiply both sides by n.
{{{150n = 180n-360}}} Add 360 to both sides.
{{{150n+360 = 180n}}} Subtract 150n from both sides.
{{{360 = 30n}}} Divide both sides by 30.
{{{12 = n}}}
This regular polygon has 12 sides and is called a "Dodecagon"