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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 $4$ sides, then find the smallest angle, in degrees.

 Jan 7, 2024
 #1
avatar+129749 
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Let d be the common difference  between the angle measures

 

Let a  be the smallest angle and  a + 56   = the largest  =  a + 3d

 

The sum of the  interior angles =  360


 

So

 

a + 3d  =  a + 56

 

3d = 56

 

d = 56/3

 

So

 

( a ) + (a + 56/3)  + (a + 2*56/3)  + (a + 3*56/3)   =  360

 

4a + (6)(56/3)  = 360

 

4a +  112  =  360

 

4a  =  360  - 112

 

4a  =  248

 

a =  248  / 4    =   62  (degrees)

 

 

cool cool cool

 Jan 7, 2024

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