A regular polygon with each interior angle 165∘165^\circ 165∘ has 24 sides.

Here’s the quick reasoning: For a regular polygon with nnn sides, each interior angle is

Interior angle=(n−2)×180n\text{Interior angle}=\frac{(n-2)\times 180}{n}Interior angle=n(n−2)×180​

Set this equal to 165:

(n−2)×180n=165\frac{(n-2)\times 180}{n}=165n(n−2)×180​=165

Multiply both sides by nnn:

180(n−2)=165n180(n-2)=165n180(n−2)=165n

Expand:

180n−360=165n180n-360=165n180n−360=165n

Bring like terms together:

180n−165n=360⇒15n=360180n-165n=360\Rightarrow 15n=360180n−165n=360⇒15n=360

So:

n=36015=24n=\frac{360}{15}=24n=15360​=24

So the polygon is a 24‑gon (regular polygon with 24 equal sides and 24 equal angles).