document.write( "Question 119079This question is from textbook Geometry for Enjoyment and Challenge
\n" ); document.write( ": 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?\r
\n" ); document.write( "\n" ); document.write( "-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. \n" ); document.write( "
Algebra.Com's Answer #87359 by bucky(2189)\"\" \"About 
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Let I represent the measure of the interior angle and E represent the measure of the exterior angle.
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\n" ); document.write( "Note that an exterior angle is formed by just extending a side of the polygon. Once you see
\n" ); document.write( "that you can also see that an interior angle and its associated exterior angle are supplementary ...
\n" ); document.write( "meaning that their measures add to 180 degrees.
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\n" ); document.write( "So, for this problem we can say that the supplementary equation is:
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\n" ); document.write( "E + I = 180
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\n" ); document.write( "The problem also tells you that the measure of each exterior angle (E) equals 25% (or 1/4) of
\n" ); document.write( "the measure of each interior angle (I). In equation form this relationship is:
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\n" ); document.write( "E = (1/4)I
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\n" ); document.write( "Substituting the right side of this equation for E in the supplementary equation results
\n" ); document.write( "in the supplementary equation becoming:
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\n" ); document.write( "(1/4)I + I = 180
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\n" ); document.write( "Get rid of the fraction by multiplying both sides of this equation (all terms) by 4 and
\n" ); document.write( "you get:
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\n" ); document.write( "I + 4I = 720
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\n" ); document.write( "The two terms on the left side sum to 5I so the equation is:
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\n" ); document.write( "5I = 720
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\n" ); document.write( "Solve for I by dividing both sides by 5 and you have:
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\n" ); document.write( "I = 720/5 = 144
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\n" ); document.write( "This means that the measure of each interior angle is 144 degrees. Since the measure of
\n" ); document.write( "the exterior angle is the supplement of the interior angle, this means that the measure of
\n" ); document.write( "each exterior angle is 180 - 144 = 36 degrees.
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\n" ); document.write( "And as you correctly said, the sum of the measures of all the exterior angles must be 360 degrees.
\n" ); document.write( "So, the number of exterior angles must be 360 divided by 36 or 10. There are, therefore,
\n" ); document.write( "10 exterior angles, and since there is one exterior angle for each side, the polygon must
\n" ); document.write( "have 10 sides. The name for a 10-sided, 10-angled polygon is \"decagon\" and for this case
\n" ); document.write( "where the measures of all the angles of the polygon are equal, you can say that it is an
\n" ); document.write( "equiangular decagon, and since all sides must also be equal, the term used for an equiangular,
\n" ); document.write( "equal sided polygon is a \"regular\" polygon. This means we can also refer to the answer as a
\n" ); document.write( "\"regular decagon.\"
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\n" ); document.write( "Hope this clears up the problem for you.
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