1. **State 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 solve for $ x $. 2. **Identify the relationship:** Since the lines are parallel and the angles are on opposite sides of the transversal but inside the parallel lines, these are alternate interior angles, which are equal. 3. **Set up the equation:** $$ 89 - 6x = 3x + 62 $$ 4. **Solve for $ x $:** Subtract 62 from both sides: $$ 89 - 6x - 62 = 3x + 62 - 62 $$ $$ 27 - 6x = 3x $$ Add $6x$ to both sides: $$ 27 - \cancel{6x} + 6x = 3x + 6x $$ $$ 27 = 9x $$ Divide both sides by 9: $$ \frac{27}{\cancel{9}} = \frac{9x}{\cancel{9}} $$ $$ 3 = x $$ 5. **Final answer:** $ x = 3 $