1. **State the problem:** Find the angle $\theta$ in triangle A-5 where the legs are 5 and unknown $\theta$, and the hypotenuse is 13. 2. **Identify the formula:** In a right triangle, the Pythagorean theorem applies: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are legs and $c$ is the hypotenuse. 3. **Apply the formula:** Given one leg $a=5$, hypotenuse $c=13$, and the other leg $b=\theta$ (unknown), we have: $$5^2 + \theta^2 = 13^2$$ 4. **Calculate:** $$25 + \theta^2 = 169$$ 5. **Isolate $\theta^2$:** $$\theta^2 = 169 - 25$$ $$\theta^2 = 144$$ 6. **Take the square root:** $$\theta = \sqrt{144}$$ $$\theta = 12$$ 7. **Interpretation:** The unknown leg $\theta$ is 12 units long. 8. **Find the angle opposite leg 5 or 12:** Using sine or cosine, for example: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ If we want the angle opposite side 5: $$\sin \alpha = \frac{5}{13}$$ $$\alpha = \arcsin\left(\frac{5}{13}\right) \approx 22.62^\circ$$ Or angle opposite side 12: $$\sin \beta = \frac{12}{13}$$ $$\beta = \arcsin\left(\frac{12}{13}\right) \approx 67.38^\circ$$ **Final answer:** The unknown leg $\theta$ is 12, and the angles opposite sides 5 and 12 are approximately $22.62^\circ$ and $67.38^\circ$ respectively.