1. **Problem:** For what value of $x$ is $\triangle ABC \sim \triangle DEF$? 2. **Step 1: Understand similarity criteria** Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. 3. **Step 2: Set corresponding angles equal** Given angles: - $\angle A = 48^\circ$ - $\angle E = 84^\circ$ - $\angle B = (x^2 - 5x)^\circ$ - $\angle D = (x^2 - 8x)^\circ$ Since $\triangle ABC \sim \triangle DEF$, corresponding angles must be equal: - $\angle A = \angle D$ or $48 = x^2 - 8x$ - $\angle B = \angle E$ or $x^2 - 5x = 84$ 4. **Step 3: Solve the equations** First equation: $$x^2 - 8x = 48$$ Rearranged: $$x^2 - 8x - 48 = 0$$ Second equation: $$x^2 - 5x = 84$$ Rearranged: $$x^2 - 5x - 84 = 0$$ 5. **Step 4: Solve quadratic equations** For $x^2 - 8x - 48 = 0$: $$x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-48)}}{2} = \frac{8 \pm \sqrt{64 + 192}}{2} = \frac{8 \pm \sqrt{256}}{2} = \frac{8 \pm 16}{2}$$ Possible values: - $x = \frac{8 + 16}{2} = 12$ - $x = \frac{8 - 16}{2} = -4$ For $x^2 - 5x - 84 = 0$: $$x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-84)}}{2} = \frac{5 \pm \sqrt{25 + 336}}{2} = \frac{5 \pm \sqrt{361}}{2} = \frac{5 \pm 19}{2}$$ Possible values: - $x = \frac{5 + 19}{2} = 12$ - $x = \frac{5 - 19}{2} = -7$ 6. **Step 5: Find common solution** The only common value for $x$ that satisfies both equations is $x = 12$. **Final answer:** $$x = 12$$