1. **Problem:** Show that segments AB and CD are proportional given points dividing them and solve for unknowns in proportional segments. 2. **Proportional segments rule:** If two segments are divided proportionally, then the ratios of the corresponding parts are equal. For example, if points E and F divide AB and CD respectively, then $ \frac{AE}{EB} = \frac{CF}{FD} $. 3. **Example D (Problem 14):** Given $ AE=4, EB=3, CF=14, FD=12 $, check if $ \frac{AE}{EB} = \frac{CF}{FD} $. Calculate: $$\frac{4}{3} \neq \frac{14}{12} = \frac{7}{6}$$ Since $ \frac{4}{3} \neq \frac{7}{6} $, segments AB and CD are not proportional in this case. 4. **Example D (Problem 15):** Given $ AE=9, EB=3, CF=6, FD=2 $, check if $ \frac{AE}{EB} = \frac{CF}{FD} $. Calculate: $$\frac{9}{3} = 3, \quad \frac{6}{2} = 3$$ Since both ratios equal 3, segments AB and CD are proportional. 5. **Example E (Problem 16):** Segments UV and WX divided proportionally by Y and Z respectively. Given $ UY=2, YV=5, ZX=40 $, find $ WZ $. Since segments are divided proportionally: $$\frac{UY}{YV} = \frac{WZ}{ZX}$$ Substitute known values: $$\frac{2}{5} = \frac{WZ}{40}$$ Cross multiply: $$2 \times 40 = 5 \times WZ$$ $$80 = 5 WZ$$ Divide both sides by 5: $$\cancel{5} \times 16 = \cancel{5} \times WZ \Rightarrow WZ = 16$$ 6. **Example E (Problem 17):** Segments AB and CD divided proportionally by points E and F. Given $ AE = x+1, EB=3, CF=7, FD=2 $, find $ x $. Set up proportion: $$\frac{AE}{EB} = \frac{CF}{FD}$$ Substitute values: $$\frac{x+1}{3} = \frac{7}{2}$$ Cross multiply: $$(x+1) \times 2 = 3 \times 7$$ $$2x + 2 = 21$$ Subtract 2 from both sides: $$2x = 19$$ Divide both sides by 2: $$\cancel{2} x = \frac{19}{\cancel{2}} \Rightarrow x = \frac{19}{2} = 9.5$$ **Final answers:** - Problem 14: AB and CD are not proportional. - Problem 15: AB and CD are proportional. - Problem 16: $ WZ = 16 $. - Problem 17: $ x = 9.5 $.