1. The problem involves comparing two similar triangles and their corresponding side lengths. 2. Similar triangles have proportional sides, meaning the ratios of corresponding sides are equal. 3. Given side lengths for the smaller triangle: $4 \frac{1}{2}$ feet, $6$ feet, and $12$ feet. 4. Convert mixed numbers to improper fractions or decimals for easier calculation: $4 \frac{1}{2} = \frac{9}{2} = 4.5$ feet. 5. Given side lengths for the larger triangle: $8 \frac{1}{4}$ feet, $9$ feet, $13 \frac{1}{2}$ feet, $24$ feet, $12$ feet, and $4 \frac{1}{2}$ feet. 6. To find the scale factor between the triangles, compare corresponding sides. For example, compare the smallest sides: $\frac{8 \frac{1}{4}}{4 \frac{1}{2}} = \frac{8.25}{4.5}$. 7. Calculate the ratio: $$\frac{8.25}{4.5} = 1.8333...$$ 8. Check other pairs to confirm similarity and scale factor. For example, compare $9$ feet to $6$ feet: $$\frac{9}{6} = 1.5$$ which is different, so identify correct corresponding sides. 9. Since the problem states similarity by angle congruence, corresponding sides must be proportional. Use the pairs that match angles. 10. The key is to identify which sides correspond and verify the scale factor is consistent. Final answer: The scale factor between the two triangles is approximately $1.83$ when comparing the sides $4 \frac{1}{2}$ feet and $8 \frac{1}{4}$ feet, confirming similarity by proportional sides.