1. **Stating the problem:** We are given two angles formed by a diagonal line crossing two parallel horizontal lines. The angles are labeled as $(15x - 2)^\circ$ and $(11x + 34)^\circ$. We need to find the value of $x$ and the measures of these angles. 2. **Understanding the relationship:** Since the two lines are parallel and the diagonal is a transversal, the given angles are alternate interior angles, which are equal. 3. **Setting up the equation:** $$ 15x - 2 = 11x + 34 $$ 4. **Solving for $x$:** Subtract $11x$ from both sides: $$ 15x - \cancel{11x} - 2 = \cancel{11x} + 34 - 11x $$ which simplifies to $$ (15x - \cancel{11x}) - 2 = 34 $$ $$ 4x - 2 = 34 $$ Add 2 to both sides: $$ 4x - 2 + 2 = 34 + 2 $$ $$ 4x = 36 $$ Divide both sides by 4: $$ \frac{4x}{\cancel{4}} = \frac{36}{\cancel{4}} $$ $$ x = 9 $$ 5. **Finding the angle measures:** Substitute $x=9$ into each angle expression: $$ 15x - 2 = 15(9) - 2 = 135 - 2 = 133^\circ $$ $$ 11x + 34 = 11(9) + 34 = 99 + 34 = 133^\circ $$ **Final answer:** - Value of $x$ is $9$. - Both angles measure $133^\circ$.