1. **State the problem:** We are given two similar triangles $\triangle ABC$ and $\triangle PQR$ with side lengths: $AB=6, BC=12, AC=9$ and $PQ=10, QR=15, PR=20$. We need to complete the proportions comparing corresponding sides and choose the correct statement about these ratios. 2. **Recall the property of similar triangles:** Corresponding sides of similar triangles are proportional. This means: $$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$$ 3. **Use the given ratios:** We are given one proportion: $$\frac{AC}{PR} = \frac{3}{5}$$ Check if this matches the actual side lengths: $$\frac{AC}{PR} = \frac{9}{20}$$ Simplify $\frac{9}{20}$ to decimal $0.45$ and $\frac{3}{5} = 0.6$, so $\frac{3}{5}$ is not equal to $\frac{9}{20}$. So the given $\frac{3}{5}$ is likely a target ratio to find. 4. **Calculate the scale factor between triangles:** Compare $AB$ and $PQ$: $$\frac{AB}{PQ} = \frac{6}{10} = \frac{3}{5}$$ This matches the given ratio $\frac{3}{5}$. 5. **Complete the proportions:** - For $\frac{AB}{?} = \frac{3}{5}$, the denominator is $PQ=10$. - For $\frac{AC}{?} = \frac{3}{5}$, the denominator is $PR=20$. - For $\frac{BC}{QR} = ?$, calculate: $$\frac{BC}{QR} = \frac{12}{15} = \frac{4}{5}$$ 6. **Final proportions:** $$\frac{AB}{PQ} = \frac{6}{10} = \frac{3}{5}$$ $$\frac{AC}{PR} = \frac{9}{20} = \frac{9}{20}$$ (does not equal $\frac{3}{5}$) $$\frac{BC}{QR} = \frac{12}{15} = \frac{4}{5}$$ 7. **Choose the correct statement:** Since the ratios are not all equal ($\frac{3}{5} \neq \frac{9}{20} \neq \frac{4}{5}$), the pairs of side lengths compared are not in the same ratio. This is because the triangles are not right triangles and the sides do not correspond proportionally. **Answer:** - $\frac{AB}{PQ} = \frac{3}{5}$ - $\frac{AC}{PR} = \frac{9}{20}$ - $\frac{BC}{QR} = \frac{4}{5}$ - Correct statement: "Each pair of side lengths compared is not in the same ratio. This is because in a proportion both ratios must be different." (Note: The first option is slightly incorrect in wording; the best fit is the first option as the ratios differ.)