1. **State the problem:** Find the value of $x$ that makes triangles $\triangle PQR$ and $\triangle TUV$ similar by using the fact that corresponding angles in similar triangles are congruent. 2. **Identify corresponding angles:** Given angles are: - $\angle Q = 2x^\circ$ - $\angle P = 40^\circ$ - $\angle V = (2x - 30)^\circ$ Since $\triangle PQR \sim \triangle TUV$, corresponding angles are equal: - $\angle Q = \angle U$ - $\angle P = \angle T$ - $\angle R = \angle V$ 3. **Use the angle sum property:** The sum of angles in a triangle is $180^\circ$. For $\triangle PQR$: $$\angle P + \angle Q + \angle R = 180^\circ$$ $$40 + 2x + \angle R = 180$$ $$\angle R = 180 - 40 - 2x = 140 - 2x$$ For $\triangle TUV$: $$\angle T + \angle U + \angle V = 180^\circ$$ Since $\angle T = 40^\circ$ (corresponding to $\angle P$), and $\angle U = 2x^\circ$ (corresponding to $\angle Q$), and $\angle V = (2x - 30)^\circ$ given. 4. **Set corresponding angles equal:** $$\angle R = \angle V$$ $$140 - 2x = 2x - 30$$ 5. **Solve for $x$:** $$140 + 30 = 2x + 2x$$ $$170 = 4x$$ $$x = \frac{170}{4} = 42.5$$ **Final answer:** $$x = 42.5$$