1. **Problem 1:** Given triangle ABC with angles $A=35^\circ$, $B=88^\circ$, and side $AC=44$ mm, find the missing angle $C$ and the missing sides. 2. **Step 1: Find angle $C$ using the angle sum property of triangles:** $$C = 180^\circ - A - B = 180^\circ - 35^\circ - 88^\circ = 57^\circ$$ 3. **Step 2: Use the Law of Sines to find missing sides $AB$ and $BC$:** The Law of Sines states: $$\frac{AB}{\sin C} = \frac{BC}{\sin A} = \frac{AC}{\sin B}$$ 4. **Step 3: Calculate side $AB$ opposite angle $C$:** $$\frac{AB}{\sin 57^\circ} = \frac{44}{\sin 88^\circ}$$ $$AB = \frac{44 \times \sin 57^\circ}{\sin 88^\circ}$$ Calculate values: $$\sin 57^\circ \approx 0.838671, \quad \sin 88^\circ \approx 0.999391$$ $$AB = \frac{44 \times 0.838671}{0.999391} \approx 36.899$$ mm 5. **Step 4: Calculate side $BC$ opposite angle $A$:** $$\frac{BC}{\sin 35^\circ} = \frac{44}{\sin 88^\circ}$$ $$BC = \frac{44 \times \sin 35^\circ}{\sin 88^\circ}$$ Calculate values: $$\sin 35^\circ \approx 0.574\quad \Rightarrow BC = \frac{44 \times 0.574}{0.999391} \approx 25.263$$ mm **Final answers for Problem 1:** - Missing angle $C = 57.000^\circ$ - Side $AB \approx 36.899$ mm - Side $BC \approx 25.263$ mm --- 2. **Problem 2:** Given triangle ABC with angles $B=51^\circ$, $A=71^\circ$, and side $BC=9.8$ cm, find the missing angle $C$ and the missing sides. 3. **Step 1: Find angle $C$:** $$C = 180^\circ - A - B = 180^\circ - 71^\circ - 51^\circ = 58^\circ$$ 4. **Step 2: Use Law of Sines:** $$\frac{AB}{\sin C} = \frac{AC}{\sin B} = \frac{BC}{\sin A}$$ 5. **Step 3: Calculate side $AB$ opposite angle $C$:** $$\frac{AB}{\sin 58^\circ} = \frac{9.8}{\sin 71^\circ}$$ $$AB = \frac{9.8 \times \sin 58^\circ}{\sin 71^\circ}$$ Calculate values: $$\sin 58^\circ \approx 0.848048, \quad \sin 71^\circ \approx 0.945519$$ $$AB = \frac{9.8 \times 0.848048}{0.945519} \approx 8.789$$ cm 6. **Step 4: Calculate side $AC$ opposite angle $B$:** $$\frac{AC}{\sin 51^\circ} = \frac{9.8}{\sin 71^\circ}$$ $$AC = \frac{9.8 \times \sin 51^\circ}{\sin 71^\circ}$$ Calculate values: $$\sin 51^\circ \approx 0.777146$$ $$AC = \frac{9.8 \times 0.777146}{0.945519} \approx 8.051$$ cm **Final answers for Problem 2:** - Missing angle $C = 58.000^\circ$ - Side $AB \approx 8.789$ cm - Side $AC \approx 8.051$ cm