To determine the number of sides in a regular polygon given that the measure of each interior angle is \(144^\circ\), we can use the formula for the measure of an interior angle of a regular polygon:
\[ \text{Interior angle} = \frac{(n-2) \times 180^\circ}{n} \]
where \(n\) is the number of sides of the polygon.
Given that the interior angle is \(144^\circ\), we set up the equation:
\[ \frac{(n-2) \times 180}{n} = 144 \]
First, we multiply both sides of the equation by \(n\) to eliminate the fraction:
\[ (n - 2) \times 180 = 144n \]
Next, we expand the left side:
\[ 180n - 360 = 144n \]
Now, we subtract \(144n\) from both sides to combine like terms:
\[ 180n - 144n - 360 = 0 \]
This simplifies to:
\[ 36n - 360 = 0 \]
Next, we add \(360\) to both sides:
\[ 36n = 360 \]
Now, divide both sides by \(36\):
\[ n = \frac{360}{36} = 10 \]
Thus, the number of sides in the polygon is \( \boxed{10} \).
To verify, we can calculate the interior angle for a 10-sided polygon (decagon):
- Calculate the total interior angles using the formula \( (n-2) \times 180 \):
\[ (10 - 2) \times 180 = 8 \times 180 = 1440^\circ \]
- Calculate the measure of each interior angle:
\[ \frac{1440}{10} = 144^\circ \]
This confirms that our calculation was correct, and the polygon is indeed a decagon with \( \boxed{10} \) sides.