1. **Problem 1:** Given triangle ABC with \(m\angle C = 60^\circ\), side \(BC = 5.5\) m, and \(m\angle A = 80^\circ\), find \(b\), \(c\), and the missing angle measure \(m\angle B\). 2. **Step 1:** Find the missing angle \(m\angle B\) using the angle sum property of triangles: $$m\angle B = 180^\circ - m\angle A - m\angle C = 180^\circ - 80^\circ - 60^\circ = 40^\circ$$ 3. **Step 2:** Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ Here, side \(a = BC = 5.5\) m opposite \(m\angle A = 80^\circ\), side \(b = AC\) opposite \(m\angle B = 40^\circ\), and side \(c = AB\) opposite \(m\angle C = 60^\circ\). 4. **Step 3:** Calculate \(b\): $$b = a \times \frac{\sin B}{\sin A} = 5.5 \times \frac{\sin 40^\circ}{\sin 80^\circ} \approx 5.5 \times \frac{0.6428}{0.9848} \approx 3.59\text{ m}$$ 5. **Step 4:** Calculate \(c\): $$c = a \times \frac{\sin C}{\sin A} = 5.5 \times \frac{\sin 60^\circ}{\sin 80^\circ} \approx 5.5 \times \frac{0.8660}{0.9848} \approx 4.84\text{ m}$$ --- 6. **Problem 2:** Given triangle ABC with \(m\angle B = 135^\circ\), \(m\angle C = 35^\circ\), and side \(BC = 3\) cm, find \(m\angle A\), \(b\), and \(c\). 7. **Step 1:** Find \(m\angle A\): $$m\angle A = 180^\circ - 135^\circ - 35^\circ = 10^\circ$$ 8. **Step 2:** Use Law of Sines with side \(a = BC = 3\) cm opposite \(m\angle A = 10^\circ\), side \(b = AC\) opposite \(m\angle B = 135^\circ\), and side \(c = AB\) opposite \(m\angle C = 35^\circ\). 9. **Step 3:** Calculate \(b\): $$b = a \times \frac{\sin B}{\sin A} = 3 \times \frac{\sin 135^\circ}{\sin 10^\circ} \approx 3 \times \frac{0.7071}{0.1736} \approx 12.21\text{ cm}$$ 10. **Step 4:** Calculate \(c\): $$c = a \times \frac{\sin C}{\sin A} = 3 \times \frac{\sin 35^\circ}{\sin 10^\circ} \approx 3 \times \frac{0.574}{0.1736} \approx 9.92\text{ cm}$$ --- 11. **Problem 3:** Given triangle ABC with \(m\angle C = 70^\circ\), side \(AC = 9.9\) km, \(m\angle B = 30^\circ\), and side \(AB = 30\) km, find \(m\angle A\), \(a = BC\), and \(b = AC\). 12. **Step 1:** Find \(m\angle A\): $$m\angle A = 180^\circ - 70^\circ - 30^\circ = 80^\circ$$ 13. **Step 2:** Use Law of Sines with side \(b = AC = 9.9\) km opposite \(m\angle B = 30^\circ\), side \(c = AB = 30\) km opposite \(m\angle C = 70^\circ\), and side \(a = BC\) opposite \(m\angle A = 80^\circ\). 14. **Step 3:** Calculate \(a\): $$a = c \times \frac{\sin A}{\sin C} = 30 \times \frac{\sin 80^\circ}{\sin 70^\circ} \approx 30 \times \frac{0.9848}{0.9397} \approx 31.43\text{ km}$$ 15. **Step 4:** Calculate \(b\): $$b = c \times \frac{\sin B}{\sin C} = 30 \times \frac{\sin 30^\circ}{\sin 70^\circ} \approx 30 \times \frac{0.5}{0.9397} \approx 15.97\text{ km}$$ --- **Final answers:** 1. \(m\angle B = 40^\circ\), \(b \approx 3.59\) m, \(c \approx 4.84\) m 2. \(m\angle A = 10^\circ\), \(b \approx 12.21\) cm, \(c \approx 9.92\) cm 3. \(m\angle A = 80^\circ\), \(a \approx 31.43\) km, \(b \approx 15.97\) km