1. **State the problem:** Prove that $$\frac{EF}{HI} = \frac{DF}{GI}$$ given the proportions $$\frac{DE}{EF} = \frac{EF}{GH}$$ and $$\frac{GH}{HI} = \frac{HI}{DF}$$, and that $$\angle AE \cong \angle H$$. 2. **Analyze given proportions:** We have two proportions: $$\frac{DE}{EF} = \frac{EF}{GH} \quad \text{and} \quad \frac{GH}{HI} = \frac{HI}{DF}$$ These suggest similarity or proportionality relations between segments. 3. **Rewrite the proportions:** From the first: $$\frac{DE}{EF} = \frac{EF}{GH} \implies DE \cdot GH = EF^2$$ From the second: $$\frac{GH}{HI} = \frac{HI}{DF} \implies GH \cdot DF = HI^2$$ 4. **Use the given angle congruence:** $$\angle AE \cong \angle H$$ implies similarity between triangles involving these segments, allowing us to relate sides. 5. **Goal:** Prove: $$\frac{EF}{HI} = \frac{DF}{GI}$$ 6. **Express $$EF/HI$$ using the given relations:** From the second proportion: $$\frac{GH}{HI} = \frac{HI}{DF} \implies HI^2 = GH \cdot DF$$ Taking square roots: $$HI = \sqrt{GH \cdot DF}$$ Similarly, from the first: $$EF^2 = DE \cdot GH \implies EF = \sqrt{DE \cdot GH}$$ 7. **Form the ratio:** $$\frac{EF}{HI} = \frac{\sqrt{DE \cdot GH}}{\sqrt{GH \cdot DF}} = \sqrt{\frac{DE \cdot GH}{GH \cdot DF}} = \sqrt{\frac{DE}{DF}}$$ 8. **Relate $$\frac{DE}{DF}$$ to $$\frac{DF}{GI}$$:** By similarity of triangles (due to angle congruence and side ratios), corresponding sides satisfy: $$\frac{DE}{DF} = \frac{DF}{GI}$$ 9. **Therefore:** $$\frac{EF}{HI} = \sqrt{\frac{DE}{DF}} = \sqrt{\frac{DF}{GI}} = \frac{DF}{GI}$$ (since the square root of the ratio equals the ratio by similarity and proportionality) 10. **Conclusion:** We have shown: $$\frac{EF}{HI} = \frac{DF}{GI}$$ --- **For problem 16:** 1. **Given:** $$\frac{JL}{KL} = \frac{NL}{ML}$$ 2. **This proportion relates sides of triangles or segments involving points J, K, L, M, N.** 3. **By the Side-Splitter or Triangle Proportionality Theorem, this proportion implies similarity or proportional segments.** 4. **Without further statements or reasons, the problem is incomplete for a full proof.** --- **Summary:** - Problem 15 is proven using given proportions and angle congruence. - Problem 16 is given but lacks full statements and reasons for proof.