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In a certain regular polygon, the measure of each interior angle is \(10\) times the measure of each exterior angle. Find the number of sides in this regular polygon.

 Jan 10, 2022
 #1
avatar+26 
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Let the measure of an exterior angle of the polygon be \(x\). Then \(x + 10x = 180^\circ\), so \(x = \left(\frac{180}{11}\right)^\circ\). The measure of an exterior angle of a regular polygon with \(\ell\) sides is equal to \(\left(\frac{360}{\ell}\right)^\circ\). Thus, \(\left(\frac{360}{\ell}\right)^\circ = \left(\frac{180}{11}\right)^\circ\), so the polygon has \(\boxed{22}\) sides.

 Jan 10, 2022
 #2
avatar+36420 
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Sum of interior anges =   ( n-2) * 180 

     now divide by n to find the measure of each interior angle    (n-2)* 180/n

        the exterior angles are supplementary   ( and exterior is 10 times bigger) 

                                                                            10*  ( 180 - (n-2)*180/n)   =   (n-2)*180/n

                                                                             1800 - 10(n-2)*180/n = (n-2)*180/n

                                                                             1800 = 11 (n-2)*180/n

                                                                             1800n = (11n-22) * 180 

                                                                              -180n = -3960 

                                                                                n = 22 sides                        

 Jan 10, 2022

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