In the image, two parallel lines are intersected by a transversal, forming two angles: $(11x - 6)^\circ$ and $(4x + 36)^\circ$.

These angles are corresponding angles, and since the lines are parallel, corresponding angles are congruent. Therefore, we can set the two angles equal to each other:

$11x - 6 = 4x + 36$

Solving for $x$:

1. Subtract $4x$ from both sides: $11x - 4x - 6 = 36$ $7x - 6 = 36$

2. Add 6 to both sides: $7x = 42$

3. Divide by 7: $x = 6$

Thus, the value of $x$ is 6.

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. What are corresponding angles and why are they congruent when the lines are parallel?
2. If one of the angles was obtuse, what could be its minimum possible value?
3. How would the calculation change if the transversal was not perpendicular to the parallel lines?
4. Can this problem be solved using another method, such as substitution?
5. How do we verify the solution once we find the value of $x$?

Tip: Always remember to verify your solution by substituting the value of $x$ back into the original expressions for the angles!