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The exterior angles of a k-sided polygon form an arithmetic sequence. The smallest and largest interior angles of the polygon are 136 and 144. What is k?

 Jul 6, 2021
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Smallest interior angle     136       exterior angle = 180 - 136 = 44 degrees

Largest interior angle       144       exterior angle = 180 -144 = 36 degrees

 

 

the exterior angles sum to 360

 

36      +     36 + d      +   36 +2d     +   36 + 3d      +    36+4d     +    36 + 5d    +    36 + 6d    +    36 + 7d    +   44   = 360

8(36) + 28d + 44 = 360

d =1

 k = 9

 

 

 

 

 

Interior angle sum:

(n-2)(180) = 7(180) =1260                 1260 / 9 = 140  (the average)            Check!

 Jul 6, 2021
edited by ElectricPavlov  Jul 6, 2021

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