To find the measure of side \( IJ \) in trapezoid \( IJKL \) given that trapezoid \( I'J'K'L' \) is a dilation of trapezoid \( IJKL \) by a scale factor of \( \frac{2}{3} \), and that side \( I'J' \) measures 44, we can use the relationship between the original and the dilated figures.
The relationship can be represented as follows:
\[ I'J' = \text{scale factor} \times IJ \] Given:
- \( I'J' = 44 \)
- scale factor = \( \frac{2}{3} \)
Substituting into the equation, we have:
\[ 44 = \frac{2}{3} \times IJ \]
To solve for \( IJ \), we can rearrange the equation:
\[ IJ = 44 \div \frac{2}{3} = 44 \times \frac{3}{2} \]
Calculating \( IJ \):
\[ IJ = 44 \times \frac{3}{2} = 44 \times 1.5 = 66 \]
Thus, the measure of side \( IJ \) is \( \boxed{66} \).