+1859 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.
+721 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.
