Calculating Polygon Sides from Angle Difference

This problem involves finding the number of sides ($n$) of a regular polygon based on the difference between its interior and exterior angles at a vertex.

Understanding Angle Formulas

For a regular polygon with $n$ sides:

• The exterior angle ($E$) is given by the formula: $E = \frac{360^\circ}{n}$
• The interior angle ($I$) is related to the exterior angle by: $I = 180^\circ - E$

Solving for the Number of Sides

We are given that the difference between the interior and exterior angles is $160^\circ$. So:

$I - E = 160^\circ$

Substitute the expression for $I$ ($I = 180^\circ - E$) into the equation:

$(180^\circ - E) - E = 160^\circ$ $180^\circ - 2E = 160^\circ$

Now, solve for the exterior angle $E$:

$2E = 180^\circ - 160^\circ$ $2E = 20^\circ$ $E = \frac{20^\circ}{2}$ $E = 10^\circ$

Finally, use the formula for the exterior angle to find the number of sides ($n$):

$E = \frac{360^\circ}{n}$ $10^\circ = \frac{360^\circ}{n}$

Rearrange the formula to solve for $n$:

$n = \frac{360^\circ}{10^\circ}$ $n = 36$

Therefore, the regular polygon has 36 sides.