1. **Stating the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are $(89 - 6x)^\circ$ and $(3x + 62)^\circ$. We need to find the value of $x$. 2. **Formula and rules:** When a transversal intersects two parallel lines, alternate interior angles are equal. Here, the two given angles are alternate interior angles, so: $$89 - 6x = 3x + 62$$ 3. **Solving the equation:** Subtract 62 from both sides: $$89 - 62 - 6x = 3x$$ $$27 - 6x = 3x$$ Add $6x$ to both sides: $$27 = 3x + 6x$$ $$27 = 9x$$ Divide both sides by 9: $$x = \frac{27}{9} = 3$$ 4. **Answer:** The value of $x$ is $3$.