1. **Problem 1: Triangle E-D-F** Given: side $ED=20$, angle $E=44^\circ$, angle $F=80^\circ$. Find missing side $DF$ and angle $D$. 2. **Find angle $D$:** Sum of angles in a triangle is $180^\circ$. $$D = 180^\circ - E - F = 180^\circ - 44^\circ - 80^\circ = 56^\circ$$ 3. **Use Law of Sines:** $$\frac{ED}{\sin F} = \frac{DF}{\sin E} = \frac{EF}{\sin D}$$ We know $ED=20$, $E=44^\circ$, $F=80^\circ$, $D=56^\circ$. 4. **Find side $DF$ opposite angle $E$:** $$DF = \frac{ED \cdot \sin E}{\sin F} = \frac{20 \cdot \sin 44^\circ}{\sin 80^\circ}$$ Calculate values: $$\sin 44^\circ \approx 0.6947, \quad \sin 80^\circ \approx 0.9848$$ $$DF = \frac{20 \times 0.6947}{0.9848} = \frac{13.894}{0.9848} \approx 14.1$$ 5. **Find side $EF$ opposite angle $D$:** $$EF = \frac{ED \cdot \sin D}{\sin F} = \frac{20 \cdot \sin 56^\circ}{\sin 80^\circ}$$ Calculate values: $$\sin 56^\circ \approx 0.8290$$ $$EF = \frac{20 \times 0.8290}{0.9848} = \frac{16.58}{0.9848} \approx 16.8$$ --- **Final answers for triangle E-D-F:** - Angle $D = 56^\circ$ - Side $DF \approx 14.1$ - Side $EF \approx 16.8$