To determine the number of sides [tex]\( n \)[/tex] in a regular polygon with each interior angle measuring [tex]\( 144^\circ \)[/tex], follow these steps:
1. Recall the formula for the interior angle of a regular [tex]\( n \)[/tex]-sided polygon:
[tex]\[
\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}
\][/tex]
2. Set the interior angle equal to [tex]\( 144^\circ \)[/tex]:
[tex]\[
144 = \frac{(n-2) \times 180}{n}
\][/tex]
3. Clear the fraction by multiplying both sides of the equation by [tex]\( n \)[/tex]:
[tex]\[
144n = (n-2) \times 180
\][/tex]
4. Distribute the [tex]\( 180 \)[/tex] on the right side:
[tex]\[
144n = 180n - 360
\][/tex]
5. Rearrange the equation to isolate [tex]\( n \)[/tex]:
[tex]\[
144n - 180n = -360
\][/tex]
[tex]\[
-36n = -360
\][/tex]
6. Divide both sides by [tex]\( -36 \)[/tex]:
[tex]\[
n = \frac{360}{36}
\][/tex]
7. Simplify the right side:
[tex]\[
n = 10
\][/tex]
So, the regular polygon has [tex]\( \boxed{10} \)[/tex] sides.
