1. **Stating the problem:** We have two triangles, DEF with sides 24, 18, and 28, and GHI with angles 45°, 58°, 60°, and one side length 70. We want to find missing sides or angles using triangle properties. 2. **Important rules:** The sum of angles in any triangle is 180°. 3. **Check angles in triangle GHI:** Given angles 45°, 58°, and 60° sum to $$45 + 58 + 60 = 163$$ which is not 180°, so there might be a typo or extra angle. Assuming 45°, 58°, and 77° (to make 180°) or focusing on given data. 4. **Using Law of Sines:** For triangle GHI, if side $GH = 70$ opposite angle $HIJ$, we can find other sides if angles are correct. 5. Since the problem is unclear about which side corresponds to which angle, we focus on triangle DEF. 6. **Triangle DEF:** Sides are 24, 18, and 28. We can find angles using Law of Cosines. 7. Law of Cosines formula: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ 8. Find angle opposite side 28: $$28^2 = 24^2 + 18^2 - 2 \times 24 \times 18 \times \cos(C)$$ $$784 = 576 + 324 - 864 \cos(C)$$ $$784 = 900 - 864 \cos(C)$$ $$864 \cos(C) = 900 - 784 = 116$$ $$\cos(C) = \frac{116}{864}$$ $$\cos(C) \approx 0.1343$$ $$C = \cos^{-1}(0.1343) \approx 82.3^\circ$$ 9. Find another angle, opposite side 24: $$24^2 = 18^2 + 28^2 - 2 \times 18 \times 28 \times \cos(B)$$ $$576 = 324 + 784 - 1008 \cos(B)$$ $$576 = 1108 - 1008 \cos(B)$$ $$1008 \cos(B) = 1108 - 576 = 532$$ $$\cos(B) = \frac{532}{1008}$$ $$\cos(B) \approx 0.5278$$ $$B = \cos^{-1}(0.5278) \approx 58.1^\circ$$ 10. Find the last angle: $$A = 180^\circ - 82.3^\circ - 58.1^\circ = 39.6^\circ$$ **Final answer:** Angles of triangle DEF are approximately $39.6^\circ$, $58.1^\circ$, and $82.3^\circ$.