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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.

 Dec 18, 2023
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In a polygon with n sides, the sum of the interior angles is (n - 2) * 180°. Since the angles form an arithmetic sequence, we can define the first angle in the sequence as a and the common difference as d.

 

1. Relate angles to sum and sequence:

 

The sum of the n interior angles can be expressed as: a + (a + d) + (a + 2d) + ... + (a + (n - 1)d) = n * a + (n - 1) * d/2.

 

We know the sum of the angles is (n - 2) * 180°: (n - 2) * 180° = n * a + (n - 1) * d/2.

 

2. Relate largest and smallest angles:

 

We are given that the difference between the largest and smallest angle is 56°: (a + (n - 1)d) - a = 56°.

 

Simplifying: (n - 1)d = 56°.

 

3. Solve for the smallest angle:

 

Combining equations 1 and 2: (n - 2) * 180° = n * a + (n - 1) * d/2 = n * a + 28n.

 

Rearranging for a: a = (n - 2) * 180° / n - 28.

 

4. Apply for given polygon (3 sides):

 

n = 3, so a = (3 - 2) * 180° / 3 - 28 = 152° - 28 = 124°.

 

Therefore, the smallest angle in the polygon is 124 degrees.

 Dec 18, 2023

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