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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.

 May 3, 2024
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Alrgiht! Not too bad of a problem!

 

First of all, if the polygon has 3 sides then there are 3 angles, pretty simply in that term. It also means that the sides add up to 180!

 

Another thing to point out is the fact that because the 3 angles make an arithmatic sequence, the difference between the smallest and the middle angle is the same as the difference between the biggest and the middle angles. Since the total is 56, we just do 56/2 = 28, so the common difference for angles ig 28.

 

Now, let's write an equation. 

 

Let s be the smallest angle. We have: \(d +(d+28) + (d+56) = 180\)

 

Combining like terms gets us \(3d + 84 = 180\), meaning \(3d = 96\) so 

 

\(d = 96/3 = 32\)

 

So the smallest angle is 32 

 

Thanks!

 May 3, 2024

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