1. The problem is to find the measure of angle $\angle E$ in a right triangle with vertices $D$, $F$, and $E$, where $\angle D$ is the right angle. 2. Given: $DF = 3$ inches, $FE = 2$ inches, and $\angle D = 90^\circ$. 3. To find $\angle E$, we use the trigonometric ratios. Since $\angle D$ is right, sides $DF$ and $FE$ are legs of the triangle, and $DE$ is the hypotenuse. 4. The formula to use is the tangent function, which relates the opposite side to the adjacent side of an angle in a right triangle: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 5. For $\angle E$, the side opposite is $DF = 3$ and the side adjacent is $FE = 2$. 6. Calculate $\tan(\angle E)$: $$\tan(\angle E) = \frac{3}{2}$$ 7. To find $\angle E$, take the arctangent (inverse tangent) of $\frac{3}{2}$: $$\angle E = \tan^{-1}\left(\frac{3}{2}\right)$$ 8. Using a calculator: $$\angle E \approx 56.3^\circ$$ 9. Therefore, the measure of $\angle E$ is approximately $56.3$ degrees to the nearest tenth.