1. **State the problem:** We need to find the measure of angle $\angle E$ in the given right triangle. 2. **Identify the triangle sides:** The triangle has a right angle at $D$. The segment $DF = 3$ inches and $FH = 4$ inches. 3. **Find the length of $DH$:** Since $DF$ and $FH$ are legs of the right triangle, use the Pythagorean theorem: $$DH = \sqrt{DF^2 + FH^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ 4. **Determine which angle is $\angle E$:** $\angle E$ is at vertex $E$, which is opposite side $DF$ and adjacent to side $FH$. 5. **Use trigonometric ratios:** To find $\angle E$, use the tangent function: $$\tan(\angle E) = \frac{\text{opposite}}{\text{adjacent}} = \frac{DF}{FH} = \frac{3}{4}$$ 6. **Calculate $\angle E$:** $$\angle E = \tan^{-1}\left(\frac{3}{4}\right)$$ 7. **Evaluate the inverse tangent:** $$\angle E \approx 36.8699^\circ$$ 8. **Round to the nearest tenth:** $$\angle E \approx 36.9^\circ$$ **Final answer:** $\boxed{36.9^\circ}$