The image shows two parallel lines (denoted by arrows indicating their parallelism) cut by a transversal. The angles formed by the intersection of the transversal with the parallel lines are labeled as $(6x + 11)^\circ$ and $77^\circ$.

Step-by-Step Solution:

1. Identify the corresponding angles: In a system of parallel lines cut by a transversal, corresponding angles are equal. The angles $(6x + 11)^\circ$ and $77^\circ$ are corresponding angles since they are on the same side of the transversal and in corresponding positions.

2. Set up the equation: Since corresponding angles are equal, we can set the two angles equal to each other: $6x + 11 = 77$

3. Solve for $x$:

   • Subtract 11 from both sides: $6x = 66$
   • Divide both sides by 6: $x = 11$

Final Answer:

$x = 11$

Would you like more details on the reasoning behind corresponding angles or any other related topics?

Here are 5 follow-up questions:

1. How do alternate interior angles relate to parallel lines and transversals?
2. What is the difference between corresponding and vertical angles?
3. How can you prove that two lines are parallel using angle relationships?
4. How are same-side interior angles useful in determining if two lines are parallel?
5. What happens to the angles if the transversal is perpendicular to the parallel lines?

Tip: Remember that the sum of interior angles on the same side of the transversal will always be supplementary (add up to 180°).