If one of the interior angles of a regular polygon is $\frac{15}{16}$ times one of the interior angles of a regular decagon, then find the number of diagonals of the polygon.

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**Solution:**

1. Calculate the interior angle of a regular decagon (10 sides):
   $$	ext{Interior angle} = \frac{(n - 2) 	imes 180^\circ}{n} = \frac{(10 - 2) 	imes 180^\circ}{10} = \frac{8 	imes 180^\circ}{10} = 144^\circ$$

2. Let the unknown polygon have $m$ sides. Its interior angle is given as:
   $$	ext{Interior angle} = \frac{15}{16} 	imes 144^\circ = 135^\circ$$

3. Using the interior angle formula for the unknown polygon:
   $$135 = \frac{(m - 2) 	imes 180}{m}$$

4. Multiply both sides by $m$:
   $$135m = 180(m - 2)$$

5. Expand and simplify:
   $$135m = 180m - 360$$

6. Bring terms involving $m$ to one side:
   $$180m - 135m = 360$$
   $$45m = 360$$

7. Solve for $m$:
   $$m = \frac{360}{45} = 8$$

8. The polygon has 8 sides, so it is an octagon.

9. Calculate the number of diagonals of an octagon:
   $$	ext{Number of diagonals} = \frac{m(m - 3)}{2} = \frac{8 	imes (8 - 3)}{2} = \frac{8 	imes 5}{2} = 20$$

**Answer:** 20

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