SOLUTION: 1. If the measure of each of the interior angles of a regular polygon is 100 more than the measure of each of the exterior angles, name the polygon? 2.) Each of the measure of

Algebra ->  Algebra  -> Polygons -> SOLUTION: 1. If the measure of each of the interior angles of a regular polygon is 100 more than the measure of each of the exterior angles, name the polygon? 2.) Each of the measure of      Log On

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Question 118697: 1. If the measure of each of the interior angles of a regular polygon is 100 more than the measure of each of the exterior angles, name the polygon?

2.) Each of the measure of the interior and exterior angles of a regular polygon are in a ration 5:1. Find the measure of an interior and an exterior angle and the name of the polygon.
please help!

Answer by Earlsdon(6103) About Me  (Show Source):
You can put this solution on YOUR website!
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%5Bi%5D%2BA%5Be%5D+=+180
b) The measure of an interior angle of a regular polygon of n sides is given by:A%5Bi%5D+=+%28n-2%29180%2Fn This works only for "regular" polygons.
In this problem, you have:
A%5Bi%5D+=+A%5Be%5D%2B100 "The interior angle is 100 degrees more than the exterior angle"
So you can write:
A%5Bi%5D%2BA%5Be%5D+=+180 Substitute A%5Bi%5D+=+A%5Be%5D%2B100
%28A%5Be%5D%2B100%29%2BA%5Be%5D+=+180 Simplify.
2A%5Be%5D%2B100+=+180 Subtract 100 from both sides.
2A%5Be%5D+=+80 Divide both sides by 2.
A%5Be%5D+=+40 The measure of an exterior angle is 40 degrees.
A%5Bi%5D+=+180-A%5Be%5D
A%5Bi%5D+=+180-40
A%5Bi%5D+=+140 The measure of an interior angle is 140 degrees.
To find the number of sides (n) in this regular polygon, use:
A%5Bi%5D+=+%28n-2%29180%2Fn Substitute A%5Bi%5D+=+140 to get:
140+=+%28n-2%29180%2Fn Simplify and solve for n, the number of sides.
140+=+%28180n-360%29%2Fn Multiply both sides by n.
140n+=+180n-360 Add 360 to both sides.
140n%2B360+=+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%5Bi%5D%2BA%5Be%5D+=+180 Substitute: A%5Bi%5D+=+5%2AA%5Be%5D
5%2AA%5Be%5D%2BA%5Be%5D+=+180 Simplify and solve for A%5Be%5D
6%2AA%5Be%5D+=+180 Divide both sides by 6.
A%5Be%5D+=+30
The exterior angle is 30 degrees.
A%5Bi%5D+=+180-A%5Be%5D Substitute A%5Be%5D+=+30
A%5Bi%5D+=+180-30
A%5Bi%5D+=+150
The interior angle is 150 degrees.
Check:
A%5Bi%5D%2FA%5Be%5D+=+5%2F1 SubstituteA%5Bi%5D+=+150 and A%5Be%5D+=+30
150%2F30+=+5%2F1 Reduce the fraction on the left side.
5%2F1+=+5%2F1
To find the number of sides (n), use:
A%5Bi%5D+=+%28n-2%29180%2Fn Substitute A%5Bi%5D+=+150 to get:
150+=+%28n-2%29180%2Fn Simplify and solve for n. Multiply both sides by n.
150n+=+180n-360 Add 360 to both sides.
150n%2B360+=+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"