Question 119079This question is from textbook Geometry for Enjoyment and Challenge
: In an equiangular polygon, the measure of each exterior angle is 25% of the measure of each interior angle. What is the name of the polygon?
-I know that the sum of the exterior angles of a polygon equals 360 degrees, but I have no idea where to go from there.
This question is from textbook Geometry for Enjoyment and Challenge
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let I represent the measure of the interior angle and E represent the measure of the exterior angle.
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Note that an exterior angle is formed by just extending a side of the polygon. Once you see
that you can also see that an interior angle and its associated exterior angle are supplementary ...
meaning that their measures add to 180 degrees.
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So, for this problem we can say that the supplementary equation is:
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E + I = 180
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The problem also tells you that the measure of each exterior angle (E) equals 25% (or 1/4) of
the measure of each interior angle (I). In equation form this relationship is:
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E = (1/4)I
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Substituting the right side of this equation for E in the supplementary equation results
in the supplementary equation becoming:
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(1/4)I + I = 180
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Get rid of the fraction by multiplying both sides of this equation (all terms) by 4 and
you get:
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I + 4I = 720
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The two terms on the left side sum to 5I so the equation is:
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5I = 720
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Solve for I by dividing both sides by 5 and you have:
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I = 720/5 = 144
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This means that the measure of each interior angle is 144 degrees. Since the measure of
the exterior angle is the supplement of the interior angle, this means that the measure of
each exterior angle is 180 - 144 = 36 degrees.
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And as you correctly said, the sum of the measures of all the exterior angles must be 360 degrees.
So, the number of exterior angles must be 360 divided by 36 or 10. There are, therefore,
10 exterior angles, and since there is one exterior angle for each side, the polygon must
have 10 sides. The name for a 10-sided, 10-angled polygon is "decagon" and for this case
where the measures of all the angles of the polygon are equal, you can say that it is an
equiangular decagon, and since all sides must also be equal, the term used for an equiangular,
equal sided polygon is a "regular" polygon. This means we can also refer to the answer as a
"regular decagon."
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Hope this clears up the problem for you.
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