1. **State the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are $ (12x - 23)^\circ $ and $ (71 - x)^\circ $. We need to solve for $ x $. 2. **Identify the relationship:** Since the lines are parallel and the angles are on the same side of the transversal, these angles are consecutive interior angles and are supplementary. This means their measures add up to 180 degrees. 3. **Write the equation:** $$ (12x - 23) + (71 - x) = 180 $$ 4. **Simplify the equation:** $$ 12x - 23 + 71 - x = 180 $$ $$ (12x - x) + (71 - 23) = 180 $$ $$ 11x + 48 = 180 $$ 5. **Isolate $ x $:** $$ 11x = 180 - 48 $$ $$ 11x = 132 $$ 6. **Solve for $ x $:** $$ x = \frac{132}{11} $$ $$ x = 12 $$ **Final answer:** $ x = 12 $