1. **State the problem:** Given that triangles $\triangle DGH$ and $\triangle DEF$ are similar, find the value of $x$. 2. **Recall the similarity rule:** Corresponding sides of similar triangles are proportional. This means: $$\frac{DG}{DE} = \frac{GH}{EF} = \frac{DH}{DF}$$ 3. **Identify corresponding sides:** - $DG = 91$ - $GH = 52$ - $DE = x + 3$ - $EF = 2x - 1$ Since $\triangle DGH \sim \triangle DEF$, the ratio of $GH$ to $EF$ equals the ratio of $DG$ to $DE$: $$\frac{GH}{EF} = \frac{DG}{DE}$$ 4. **Set up the proportion:** $$\frac{52}{2x - 1} = \frac{91}{x + 3}$$ 5. **Cross multiply:** $$52(x + 3) = 91(2x - 1)$$ 6. **Expand both sides:** $$52x + 156 = 182x - 91$$ 7. **Bring all terms to one side:** $$52x + 156 - 182x + 91 = 0$$ Simplify: $$\cancel{52x} + 156 - \cancel{182x} + 91 = 0$$ $$-130x + 247 = 0$$ 8. **Solve for $x$:** $$-130x = -247$$ $$x = \frac{-247}{-130}$$ $$x = \frac{247}{130}$$ 9. **Final answer:** $$x = \frac{247}{130} \approx 1.9$$ This is the value of $x$ that satisfies the similarity condition.