1. **State the problem:** Solve the right triangle with given angle $A=32.5^\circ$, side $b=34$, and angle $B=57.5^\circ$. Find side $a$, side $c$, and verify angles. 2. **Recall the triangle angle sum rule:** The sum of angles in a triangle is $180^\circ$. Since $C$ is a right angle, $C=90^\circ$. 3. **Check angles:** $A + B + C = 32.5^\circ + 57.5^\circ + 90^\circ = 180^\circ$, which is correct. 4. **Use the Law of Sines:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 5. **Calculate side $a$:** $$a = b \times \frac{\sin A}{\sin B} = 34 \times \frac{\sin 32.5^\circ}{\sin 57.5^\circ}$$ 6. **Calculate the sine values:** $$\sin 32.5^\circ \approx 0.5373, \quad \sin 57.5^\circ \approx 0.8434$$ 7. **Substitute and simplify:** $$a = 34 \times \frac{0.5373}{0.8434} = 34 \times 0.6371 = 21.66$$ 8. **Calculate side $c$ using Pythagoras theorem:** $$c = \sqrt{a^2 + b^2} = \sqrt{21.66^2 + 34^2}$$ 9. **Calculate squares:** $$21.66^2 = 469.16, \quad 34^2 = 1156$$ 10. **Sum and square root:** $$c = \sqrt{469.16 + 1156} = \sqrt{1625.16} = 40.31$$ 11. **Round answers:** - $a \approx 21.66$ (nearest hundredth) - $c \approx 40.31$ (nearest hundredth) - Angles $A=32.5^\circ$, $B=57.5^\circ$, $C=90^\circ$ (nearest tenth) **Final answers:** $$a = 21.66, \quad c = 40.31, \quad A = 32.5^\circ, \quad B = 57.5^\circ, \quad C = 90^\circ$$