1. **State the problem:** Calculate the measure of angle $\theta$ to the nearest degree using the given triangle and trigonometric relationships. 2. **Recall the tangent function:** For a right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. 3. **Given:** $\tan 35^\circ = \frac{\text{opp}}{\text{adj}}$, and $x = 10 \times \tan 33^\circ = 12.5$ (approximate). 4. **Use the Law of Cosines:** $$a^2 = 12.6^2 + 18.3^2 - 2 \times 18.3 \times 12.6 \times \cos 55^\circ$$ Calculate each term: $$12.6^2 = 158.76$$ $$18.3^2 = 334.89$$ $$2 \times 18.3 \times 12.6 = 460.68$$ $$\cos 55^\circ \approx 0.5736$$ So, $$a^2 = 158.76 + 334.89 - 460.68 \times 0.5736 = 493.65 - 264.68 = 228.97$$ 5. **Find $a$:** $$a = \sqrt{228.97} \approx 15.1$$ 6. **Calculate $\theta$ using tangent inverse:** Since $\tan 35^\circ = \frac{\text{opp}}{\text{adj}}$, and $x = 12.5$, the angle $\theta$ is approximately $35^\circ$. 7. **Final answer:** $$\boxed{\theta = 34^\circ}$$ (to the nearest degree as given).