To solve for the number of sides [tex]\( n \)[/tex] in a regular polygon where each interior angle is [tex]\( 144^\circ \)[/tex], we need to use the formula for the interior angle of a regular polygon.
### Step-by-Step Solution:
1. Formula for the Interior Angle:
The formula for each interior angle [tex]\( A \)[/tex] of a regular polygon with [tex]\( n \)[/tex] sides is:
[tex]\[
A = \frac{(n - 2) \cdot 180^\circ}{n}
\][/tex]
2. Set Up the Equation:
Given that each interior angle [tex]\( A \)[/tex] is [tex]\( 144^\circ \)[/tex], we substitute this value into the formula:
[tex]\[
144 = \frac{(n - 2) \cdot 180}{n}
\][/tex]
3. Solve for [tex]\( n \)[/tex]:
We now solve the equation step-by-step:
[tex]\[
144 = \frac{180n - 360}{n}
\][/tex]
Multiply both sides by [tex]\( n \)[/tex] to clear the fraction:
[tex]\[
144n = 180n - 360
\][/tex]
Move all terms involving [tex]\( n \)[/tex] to one side of the equation:
[tex]\[
144n - 180n = -360
\][/tex]
Simplify the equation:
[tex]\[
-36n = -360
\][/tex]
Divide both sides by [tex]\(-36\)[/tex]:
[tex]\[
n = \frac{360}{36}
\][/tex]
Therefore:
[tex]\[
n = 10
\][/tex]
Thus, the number of sides [tex]\( n \)[/tex] in the regular polygon is [tex]\( 10 \)[/tex].
So the answer is:
[tex]\[
\boxed{10}
\][/tex]
