1. **State the problem:** We have triangle ABC with angles $\angle A = 35^\circ$, $\angle B = 88^\circ$, and side $AC = 44$ mm. We need to find $m\angle C$, side $c$ (opposite $\angle C$), and side $a$ (opposite $\angle A$). 2. **Use the triangle angle sum rule:** The sum of angles in a triangle is $180^\circ$. $$m\angle C = 180^\circ - m\angle A - m\angle B = 180^\circ - 35^\circ - 88^\circ = 57^\circ$$ 3. **Use the Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ We know side $b = AC = 44$ mm opposite $\angle B = 88^\circ$. 4. **Find side $c$ opposite $\angle C$:** $$\frac{c}{\sin 57^\circ} = \frac{44}{\sin 88^\circ}$$ Calculate intermediate step: $$c = \frac{44 \times \sin 57^\circ}{\sin 88^\circ}$$ Since $\sin 88^\circ \approx 0.999$, we can write: $$c \approx 44 \times \sin 57^\circ$$ Calculate $\sin 57^\circ \approx 0.8387$: $$c \approx 44 \times 0.8387 = 36.9 \text{ mm}$$ 5. **Find side $a$ opposite $\angle A$:** $$\frac{a}{\sin 35^\circ} = \frac{44}{\sin 88^\circ}$$ Calculate intermediate step: $$a = \frac{44 \times \sin 35^\circ}{\sin 88^\circ}$$ Since $\sin 88^\circ \approx 0.999$, we can write: $$a \approx 44 \times \sin 35^\circ$$ Calculate $\sin 35^\circ \approx 0.574$: $$a \approx 44 \times 0.574 = 25.3 \text{ mm}$$ **Final answers:** $$m\angle C = 57^\circ$$ $$c \approx 36.9 \text{ mm}$$ $$a \approx 25.3 \text{ mm}$$