1. **State the problem:** We have two triangles, FGH and FIJ, similar by SAS similarity theorem. Given sides FG = 1, GH = 2 in the smaller triangle, and IJ = 2.5 in the larger triangle, we need to find the length of side HJ. 2. **Recall the SAS similarity theorem:** If two triangles have two sides in proportion and the included angle equal, then the triangles are similar. Corresponding sides are proportional. 3. **Set up the proportion:** Since FG corresponds to FI and GH corresponds to HJ, and we know GH and IJ, we can write the ratio: $$\frac{FG}{FI} = \frac{GH}{HJ} = \frac{1}{FI} = \frac{2}{HJ}$$ We also know IJ corresponds to GH, so: $$\frac{GH}{IJ} = \frac{2}{2.5} = \frac{4}{5}$$ 4. **Find the scale factor:** The scale factor from smaller to larger triangle is $\frac{IJ}{GH} = \frac{2.5}{2} = \frac{5}{4}$. 5. **Calculate HJ:** Since HJ corresponds to FI, and FI corresponds to FG scaled by $\frac{5}{4}$, then: $$FI = FG \times \frac{5}{4} = 1 \times \frac{5}{4} = \frac{5}{4}$$ 6. **Use the proportion to find HJ:** Using the ratio: $$\frac{GH}{HJ} = \frac{FG}{FI}$$ Substitute known values: $$\frac{2}{HJ} = \frac{1}{\frac{5}{4}} = \frac{4}{5}$$ Cross multiply: $$2 \times 5 = 4 \times HJ$$ $$10 = 4 \times HJ$$ Divide both sides by 4: $$HJ = \frac{10}{4} = \frac{5}{2} = 2.5$$ 7. **Final answer:** The length of side HJ is $2.5$.