To find the number of sides in a regular polygon when the interior angle is given as $175^\circ$, we can use the formula for the interior angle of a regular polygon:

$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$

where $n$ is the number of sides.

Step-by-Step Solution

1. Set up the equation: Since the interior angle is given as $175^\circ$, we have: $\frac{(n - 2) \times 180}{n} = 175$

2. Multiply both sides by $n$ to eliminate the denominator: $(n - 2) \times 180 = 175n$

3. Expand and rearrange terms: $180n - 360 = 175n$ $180n - 175n = 360$ $5n = 360$

4. Solve for $n$: $n = \frac{360}{5} = 72$

Conclusion

The polygon has 72 sides.

Would you like a deeper explanation of any step, or have further questions?

Here are five related questions to explore:

1. What is the formula for the exterior angle of a regular polygon?
2. How does the number of sides affect the measure of each interior angle in a polygon?
3. What would happen if the interior angle were slightly different, like $170^\circ$?
4. Can a regular polygon have an interior angle greater than $180^\circ$?
5. What are the interior and exterior angles of a decagon (10-sided polygon)?

Tip: For regular polygons, the sum of the exterior angles always adds up to $360^\circ$, regardless of the number of sides.