Sure! Let's work through the problem together step-by-step.
1. Understand the Relationship Between Interior Angles and the Number of Sides:
The formula for the measure of an interior angle of a regular polygon (where all angles and sides are equal) is given by:
[tex]\[
\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}
\][/tex]
Here, [tex]\( n \)[/tex] represents the number of sides of the polygon.
2. Set Up the Equation with the Given Interior Angle:
According to the problem, the interior angle of the polygon is [tex]\( 144^\circ \)[/tex]. So we set up the equation:
[tex]\[
144^\circ = \frac{(n-2) \times 180^\circ}{n}
\][/tex]
3. Solve for [tex]\( n \)[/tex]:
To find the number of sides [tex]\( n \)[/tex], we need to isolate [tex]\( n \)[/tex]. We can do this by multiplying both sides of the equation by [tex]\( n \)[/tex] to get rid of the fraction:
[tex]\[
144n = (n-2) \times 180
\][/tex]
Next, distribute the [tex]\( 180 \)[/tex] on the right-hand side:
[tex]\[
144n = 180n - 360
\][/tex]
4. Isolate [tex]\( n \)[/tex] by Combining Like Terms:
Subtract [tex]\( 180n \)[/tex] from both sides to isolate the [tex]\( n \)[/tex] term:
[tex]\[
144n - 180n = -360
\][/tex]
Simplify the left-hand side:
[tex]\[
-36n = -360
\][/tex]
5. Solve for [tex]\( n \)[/tex]:
Divide both sides by [tex]\(-36\)[/tex]:
[tex]\[
n = \frac{360}{36}
\][/tex]
Simplify the right-hand side:
[tex]\[
n = 10
\][/tex]
Therefore, the number of sides of the polygon is [tex]\( 10 \)[/tex].
So, a regular polygon with each interior angle measuring [tex]\( 144^\circ \)[/tex] has [tex]\( 10 \)[/tex] sides.
