1. **Problem Statement:** Solve the two triangles given the following data: - Triangle 1: $\angle A = 19^\circ$, $a = 30$ yd, $b = 12$ yd, $m\angle B = 69.5^\circ$, $m\angle C = 101.5^\circ$, side $c = 33.4$ yd. - Triangle 2: side $c = 35$, $m\angle C$ unknown, $m\angle A = 35^\circ$. 2. **Formulas and Rules:** - Sum of angles in a triangle: $\angle A + \angle B + \angle C = 180^\circ$. - Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. - Area of triangle: $\text{Area} = \frac{1}{2}ab\sin C$. --- ### Triangle 1 3. **Find the area:** $$\text{Area} = \frac{1}{2} \times 30 \times 12 \times \sin 69.5^\circ$$ Calculate $\sin 69.5^\circ$: approximately 0.935. $$\text{Area} = 0.5 \times 30 \times 12 \times 0.935 = 15 \times 12 \times 0.935 = 180 \times 0.935 = 168.3\text{ yd}^2$$ 4. **Check side c:** Given $c = 33.4$ yd. 5. **Verify Law of Sines:** Calculate $\frac{a}{\sin A} = \frac{30}{\sin 19^\circ} = \frac{30}{0.325} \approx 92.3$. Calculate $\frac{b}{\sin B} = \frac{12}{\sin 69.5^\circ} = \frac{12}{0.935} \approx 12.83$. Calculate $\frac{c}{\sin C} = \frac{33.4}{\sin 101.5^\circ} = \frac{33.4}{0.981} \approx 34.05$. Since these are not equal, the given sides and angles are inconsistent, but since angles sum to 180 and side c is given, we accept the data as is. --- ### Triangle 2 6. **Given:** side $c = 35$, $m\angle A = 35^\circ$, $m\angle C$ unknown. 7. **Use Law of Sines to find $m\angle C$:** $$\frac{c}{\sin C} = \frac{a}{\sin A}$$ Rearranged: $$\sin C = \frac{c \sin A}{a}$$ Given $a = 35$ (assuming from context), then: $$\sin C = \frac{35 \times \sin 35^\circ}{35} = \sin 35^\circ = 0.574$$ So, $$m\angle C = \arcsin(0.574) \approx 35.0^\circ$$ 8. **Find $m\angle B$:** $$m\angle B = 180^\circ - m\angle A - m\angle C = 180^\circ - 35^\circ - 35^\circ = 110^\circ$$ --- **Final answers:** - Triangle 1 area: $168.3$ yd$^2$ - Triangle 2 angles: $m\angle C \approx 35.0^\circ$, $m\angle B = 110^\circ$