1. **State the problem:** We have a right triangle DEF with a right angle at F, side DF = 4 units, and angle D = 63°. 2. **Find the missing parts:** We need to find DE, EF, and the measure of angle E. 3. **Use triangle angle sum rule:** The sum of angles in a triangle is 180°. $$m\angle D + m\angle E + m\angle F = 180^\circ$$ Since $m\angle F = 90^\circ$ and $m\angle D = 63^\circ$, then $$m\angle E = 180^\circ - 90^\circ - 63^\circ = 27^\circ$$ 4. **Use trigonometric ratios:** In right triangle DEF, with right angle at F, side DF is adjacent to angle D, EF is opposite angle D, and DE is the hypotenuse. 5. **Find DE (hypotenuse) using cosine:** $$\cos(63^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{DF}{DE} = \frac{4}{DE}$$ Rearranged: $$DE = \frac{4}{\cos(63^\circ)}$$ Calculate: $$DE = \frac{4}{0.4540} \approx 8.81$$ 6. **Find EF (opposite side) using sine:** $$\sin(63^\circ) = \frac{EF}{DE}$$ Rearranged: $$EF = DE \times \sin(63^\circ) = 8.81 \times 0.8910 \approx 7.85$$ **Final answers:** $$DE \approx 8.81$$ $$EF \approx 7.85$$ $$m\angle E = 27^\circ$$