To solve this problem, let's analyze the given information.

1. We have two parallel lines with two angles marked in between.
2. The angle labeled $66^\circ$ is an interior angle on one side of the transversal line.
3. The angles $2x^\circ$ and $4x^\circ$ are likely part of a triangle formed between the lines and the transversal.

Step-by-Step Solution:

1. Identify the Triangle: The angles $2x^\circ$ and $4x^\circ$ are angles within a triangle that also includes the angle formed by $66^\circ$ due to the parallel lines.

2. Use the Sum of Angles in a Triangle: The sum of the angles in any triangle is $180^\circ$.

   So, $2x + 4x + 66 = 180$

3. Simplify the Equation: Combine like terms: $6x + 66 = 180$

4. Solve for $x$:

   • Subtract $66$ from both sides: $6x = 114$
   • Divide by $6$: $x = 19$

Conclusion:

The value of $x$ is $19^\circ$.

Would you like more details, or do you have any questions?

Here are five related questions to deepen your understanding:

1. What if the angle labeled $66^\circ$ were different? How would that change the calculation?
2. How do we know that the triangle sum rule applies in this scenario?
3. What if the lines weren’t parallel? Would $x$ still be $19^\circ$?
4. How could you verify the solution by rechecking the angle properties?
5. Are there other geometric properties you could use to solve this problem?

Tip: When dealing with parallel lines and transversals, remember that alternate interior angles are equal, and the sum of angles in a triangle always equals $180^\circ$.