1. **Problem:** Investigate if $XY \parallel PQ$ given points and lengths. 2. **Step 1:** To check if two lines are parallel, we compare the ratios of corresponding segments. If the ratios are equal, the lines are parallel. 3. **Step 2:** Given lengths for $XY$ and $PQ$ are not explicitly stated, but assuming the points and lengths are as follows: $|XY|=7.5$, $|PQ|=8$, $|RX|=9$, $|YQ|=11$ (assuming these are segments related to the problem). 4. **Step 3:** Calculate the ratio $\frac{|XY|}{|PQ|} = \frac{7.5}{8} = 0.9375$. 5. **Step 4:** Calculate the ratio $\frac{|RX|}{|YQ|} = \frac{9}{11} = 0.8181$. 6. **Step 5:** Since $0.9375 \neq 0.8181$, the ratios are not equal, so $XY \not\parallel PQ$. 1. **Problem:** Investigate if $PQ \parallel AC$ given points and lengths. 2. **Step 1:** Given lengths: $|PQ|=24$, $|AC|=8$, $|PB|=9$, $|QC|=12$ (assuming these are the segments). 3. **Step 2:** Calculate the ratio $\frac{|PQ|}{|AC|} = \frac{24}{8} = 3$. 4. **Step 3:** Calculate the ratio $\frac{|PB|}{|QC|} = \frac{9}{12} = 0.75$. 5. **Step 4:** Since $3 \neq 0.75$, the ratios are not equal, so $PQ \not\parallel AC$. **Final answers:** - $XY \not\parallel PQ$ - $PQ \not\parallel AC$