1. **Problem:** Given triangle with angles $m\angle B = 135^\circ$, $m\angle C = 30^\circ$, side $b = 35.2$ mm, find sides $a$, $c$ and triangle type. 2. **Formula:** Use Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 3. **Find $m\angle A$:** Sum of angles in triangle is $180^\circ$, so $$A = 180^\circ - 135^\circ - 30^\circ = 15^\circ$$ 4. **Calculate $a$:** $$a = b \times \frac{\sin A}{\sin B} = 35.2 \times \frac{\sin 15^\circ}{\sin 135^\circ} = 35.2 \times \frac{0.2588}{0.7071} \approx 12.88\text{ mm}$$ 5. **Calculate $c$:** $$c = b \times \frac{\sin C}{\sin B} = 35.2 \times \frac{\sin 30^\circ}{\sin 135^\circ} = 35.2 \times \frac{0.5}{0.7071} \approx 24.87\text{ mm}$$ 6. **Triangle type:** Since one angle is $135^\circ > 90^\circ$, triangle is **obtuse**. --- 1. **Problem:** Triangle with $m\angle C = 65^\circ$, $m\angle A = 65^\circ$, side $c = 13.1$ cm, find $a$, $b$, and type. 2. **Find $m\angle B$:** $$B = 180^\circ - 65^\circ - 65^\circ = 50^\circ$$ 3. **Use Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 4. **Calculate $a$:** $$a = c \times \frac{\sin A}{\sin C} = 13.1 \times \frac{\sin 65^\circ}{\sin 65^\circ} = 13.1\text{ cm}$$ 5. **Calculate $b$:** $$b = c \times \frac{\sin B}{\sin C} = 13.1 \times \frac{\sin 50^\circ}{\sin 65^\circ} = 13.1 \times \frac{0.7660}{0.9063} \approx 11.07\text{ cm}$$ 6. **Triangle type:** All angles less than $90^\circ$, so **acute**. --- 1. **Problem:** Triangle with $m\angle A = 60^\circ$, $m\angle B = 60^\circ$, $m\angle C = 80^\circ$, side $c = 10.0$ m, find $a$, $b$, and type. 2. **Find $m\angle B$:** Given as $60^\circ$. 3. **Use Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 4. **Calculate $a$:** $$a = c \times \frac{\sin A}{\sin C} = 10.0 \times \frac{\sin 60^\circ}{\sin 80^\circ} = 10.0 \times \frac{0.8660}{0.9848} \approx 8.79\text{ m}$$ 5. **Calculate $b$:** $$b = c \times \frac{\sin B}{\sin C} = 10.0 \times \frac{\sin 60^\circ}{\sin 80^\circ} = 8.79\text{ m}$$ 6. **Triangle type:** All angles less than $90^\circ$, so **acute**. --- 1. **Problem:** Triangle with $m\angle B = 32^\circ$, $m\angle C = 32^\circ$, $m\angle A = 99^\circ$, side $c = 75$ km, find $a$, $b$, and type. 2. **Use Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 3. **Calculate $a$:** $$a = c \times \frac{\sin A}{\sin C} = 75 \times \frac{\sin 99^\circ}{\sin 32^\circ} = 75 \times \frac{0.9877}{0.5299} \approx 139.8\text{ km}$$ 4. **Calculate $b$:** $$b = c \times \frac{\sin B}{\sin C} = 75 \times \frac{\sin 32^\circ}{\sin 32^\circ} = 75\text{ km}$$ 5. **Triangle type:** Angle $99^\circ > 90^\circ$, so **obtuse**. --- 1. **Problem:** Triangle with $m\angle B = 60^\circ$, $m\angle A = 55^\circ$, $m\angle C = 60^\circ$, side $c = 120.5$ m, find $a$, $b$, and type. 2. **Find $m\angle B$:** Given as $60^\circ$. 3. **Use Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 4. **Calculate $a$:** $$a = c \times \frac{\sin A}{\sin C} = 120.5 \times \frac{\sin 55^\circ}{\sin 60^\circ} = 120.5 \times \frac{0.8192}{0.8660} \approx 113.9\text{ m}$$ 5. **Calculate $b$:** $$b = c \times \frac{\sin B}{\sin C} = 120.5 \times \frac{\sin 60^\circ}{\sin 60^\circ} = 120.5\text{ m}$$ 6. **Triangle type:** All angles less than $90^\circ$, so **acute**. --- 1. **Problem:** Triangle with $m\angle A = 102^\circ$, $m\angle C = 23^\circ$, side $c = 600$ km, find $a$, $b$, and type. 2. **Find $m\angle B$:** $$B = 180^\circ - 102^\circ - 23^\circ = 55^\circ$$ 3. **Use Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 4. **Calculate $a$:** $$a = c \times \frac{\sin A}{\sin C} = 600 \times \frac{\sin 102^\circ}{\sin 23^\circ} = 600 \times \frac{0.9781}{0.3907} \approx 1501.5\text{ km}$$ 5. **Calculate $b$:** $$b = c \times \frac{\sin B}{\sin C} = 600 \times \frac{\sin 55^\circ}{\sin 23^\circ} = 600 \times \frac{0.8192}{0.3907} \approx 1258.5\text{ km}$$ 6. **Triangle type:** Angle $102^\circ > 90^\circ$, so **obtuse**.