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The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $56^\circ$. If the polygon has $3$ sides, then find the smallest angle, in degrees.

 Oct 5, 2024
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The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $56^\circ$. If the polygon has $3$ sides, then find the smallest angle, in degrees.    

 

Call the three angles a, b, and c, in order of size.         

 

The problem states c – a = 56         ==>>   c = a + 56    

 

The angles are in an arithmetic

sequence, so b – a = (1/2)(56)         ==>>   b = a + 28    

 

The sum of the three angles of a    

triangle is 180, so a + b + c = 180    ==>>  (a) + (a + 28) + (a + 56)  =  180    

                                                                                             3a + 84  =  180    

                                                                                                     3a  =  96    

                                                                                                       a  =  32    

 

check answer    

 

                           a + b + c = 180   

32 + (32 + 28) + (32 + 56) = 180    

                                   180 = 180      ==>> True   

.    

 Oct 5, 2024

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