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avatar+1234 

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.

 Jun 13, 2024
 #1
avatar+1248 
+1

If a polygon has 3 sides, then it has to be a triangle. 

We can solve this problem by setting some variables. Let's set the smallest angle degrees as n. 

 

Now, since the angles from an arithmetic sequence, each angle is increasing an equal amount. 

This means if the first and last angle are 56 degrees apart, then the middle angle is \(56/2=23\) degrees bigger than n. 

 

Thus, we can write the equation \(n+(n+23)+(n+56)=180\)

 

Solving this equation, we get

\(3n+79=180\\ 3n=101\\ n=101/3\)

 

So our final answer is 101/3

 

This is not a whole number, but it does work. 

 

Thanks! :)

 Jun 13, 2024
 #2
avatar+129732 
+1

Let x be the smallest angle

 

Let  x + 2d   = the largest

 

(x + 2d)  - x  = 56

 

2d = 56

 

d =28

 

So

 

x + (x + d)  + (x + 2d)   =180

 

3x + 3d   =180

 

x + d  = 60

 

x + 28 = 60

 

x =60 - 28 =  32  = the smallest angle

 

cool cool cool

 Jun 14, 2024

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