# SOLUTION: Please help me answer or solve this question: How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle? It wo

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 Click here to see ALL problems on Polygons Question 109127: Please help me answer or solve this question: How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle? It would be very helpful if you could explain the process that you took to solve this problem. Thanks in advance, Christine Found 2 solutions by stanbon, Edwin McCravy:Answer by stanbon(57328)   (Show Source): You can put this solution on YOUR website!How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle? ------------- Draw the figure of an exterior angle and its corresponding interior angle. Notice that they are supplementary. --------- Let the exterior angle have measure "x"; The interior angle has measure "8x". EQUATION: x + 8x = 180 9x = 180 x = 20 degrees (measure of one of the exterior angles. -------------- Fact: The sume of the exterior angles of a polygon is 360 degrees. -------------- EQUATION: # of sides = 360/20 = 18 sides ===================== Cheers, Stan H. Answer by Edwin McCravy(8908)   (Show Source): You can put this solution on YOUR website! Please help me answer or solve this question: How many sides does a polygon have if the measure of each interior angle is 8 times the measure of each exterior angle? It would be very helpful if you could explain the process that you took to solve this problem. Thanks in advance, Christine ``` The drawing below shows the bottom right-hand corner of the polygon: The small acute angle marked x is an exterior angle of the polygon and the large obtuse angle marked 8x is an interior angle. It measures 8x because we are told that each interior angle is 8 times the measure of each exterior angle. Since the interior and exterior angle at the same vertex of the polygon are supplementary, we can equate their sum to 180°. INTERIOR ANGLE + EXTERIOR ANGLE = 180° 8x + x = 180° 9x = 180° x = 20° This means each exterior angle is 20° and each interior angle is 180°-20° or 160°. This tells us that the polygon is a regular polygon because all the interior angles are equal in measure. Now we only need to know that the theorem that says: The sum of all the exterior angles of any polynomial is always 360°. Let the number of sides of this polygon be N. Since we know that this is an N-sided regular polygon, the sum of the exterior angles is N times 20° or 20N degrees. So 20N degrees = 360 degrees 20N = 360 N = 18 sides So it is an 18-sided regular polygon. Edwin```