1. **State the problem:** We are given a triangle with some sides and angles labeled inconsistently. We need to solve the triangle, meaning find all unknown sides and angles. Given data (interpreted reasonably): - Side $c = 35$ km (or possibly 21 km, but we will consider both cases separately as two triangles). - Angle $A = 29^\circ$ (correcting the unit from km to degrees). - Angle $B = 97^\circ$. - Additional notes: $m\angle A = 62^\circ$ and $m\angle C = 24^\circ$ (likely for the second triangle). We will solve two triangles: **Triangle 1:** $c=35$ km, $A=29^\circ$, $B=97^\circ$ **Triangle 2:** $c=21$ km, $A=62^\circ$, $C=24^\circ$ --- 2. **Triangle 1: Solve for $a$, $b$, and $C$** - Sum of angles in a triangle: $$A + B + C = 180^\circ$$ - Calculate $C$: $$C = 180^\circ - 29^\circ - 97^\circ = 54^\circ$$ - Use Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ - Calculate $a$: $$a = c \times \frac{\sin A}{\sin C} = 35 \times \frac{\sin 29^\circ}{\sin 54^\circ}$$ - Calculate $b$: $$b = c \times \frac{\sin B}{\sin C} = 35 \times \frac{\sin 97^\circ}{\sin 54^\circ}$$ - Calculate numerical values: $$\sin 29^\circ \approx 0.4848, \sin 54^\circ \approx 0.8090, \sin 97^\circ \approx 0.9925$$ $$a = 35 \times \frac{0.4848}{0.8090} \approx 35 \times 0.5997 = 20.99$$ $$b = 35 \times \frac{0.9925}{0.8090} \approx 35 \times 1.227 = 42.95$$ --- 3. **Triangle 2: Solve for $a$, $b$, and $B$** - Given $c=21$ km, $A=62^\circ$, $C=24^\circ$ - Calculate $B$: $$B = 180^\circ - 62^\circ - 24^\circ = 94^\circ$$ - Use Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ - Calculate $a$: $$a = c \times \frac{\sin A}{\sin C} = 21 \times \frac{\sin 62^\circ}{\sin 24^\circ}$$ - Calculate $b$: $$b = c \times \frac{\sin B}{\sin C} = 21 \times \frac{\sin 94^\circ}{\sin 24^\circ}$$ - Calculate numerical values: $$\sin 62^\circ \approx 0.8829, \sin 24^\circ \approx 0.4067, \sin 94^\circ \approx 0.9986$$ $$a = 21 \times \frac{0.8829}{0.4067} \approx 21 \times 2.171 = 45.59$$ $$b = 21 \times \frac{0.9986}{0.4067} \approx 21 \times 2.456 = 51.58$$ --- **Final answers:** - Triangle 1: - $A=29^\circ$, $B=97^\circ$, $C=54^\circ$ - $a \approx 20.99$ km - $b \approx 42.95$ km - $c=35$ km - Triangle 2: - $A=62^\circ$, $B=94^\circ$, $C=24^\circ$ - $a \approx 45.59$ km - $b \approx 51.58$ km - $c=21$ km