1. **State the problem:** We have a triangle with angles 30°, 105°, and an unknown angle, and sides opposite 105° labeled $\sqrt{2}$, opposite 30° labeled $a$, and the base side labeled $b$. We want to find $a$ and $b$. 2. **Find the unknown angle:** The sum of angles in a triangle is 180°. $$\text{Unknown angle} = 180^\circ - 30^\circ - 105^\circ = 45^\circ$$ 3. **Use the Law of Sines:** The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. 4. **Assign sides and angles:** - Side opposite 30° is $a$ - Side opposite 105° is $\sqrt{2}$ - Side opposite 45° is $b$ 5. **Set up ratios:** $$\frac{a}{\sin 30^\circ} = \frac{\sqrt{2}}{\sin 105^\circ} = \frac{b}{\sin 45^\circ}$$ 6. **Calculate $a$:** $$a = \frac{\sin 30^\circ}{\sin 105^\circ} \times \sqrt{2}$$ Since $\sin 30^\circ = 0.5$ and $\sin 105^\circ = \sin(180^\circ - 75^\circ) = \sin 75^\circ \approx 0.9659$, we get: $$a = \frac{0.5}{0.9659} \times \sqrt{2} \approx 0.5176 \times 1.4142 = 0.732$$ 7. **Calculate $b$:** $$b = \frac{\sin 45^\circ}{\sin 105^\circ} \times \sqrt{2}$$ Since $\sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.7071$, we get: $$b = \frac{0.7071}{0.9659} \times \sqrt{2} \approx 0.732 \times 1.4142 = 1.035$$ **Final answers:** $$a \approx 0.732$$ $$b \approx 1.035$$