To solve for the angle $x$ in the image, we can use the following steps:

1. Identify the angles:

   • The angle marked $78^\circ$ is given.
   • The angle $\alpha + 26^\circ$ is also given.
   • The angle $\alpha$ is marked adjacent to $x$.

2. Observe the relationship between angles:

   • The angle $78^\circ$ and $\alpha + 26^\circ$ are alternate interior angles, which means they are equal. Hence: $\alpha + 26^\circ = 78^\circ$ Solving for $\alpha$: $\alpha = 78^\circ - 26^\circ = 52^\circ$

3. Find the value of $x$:

   • The angle $x$ and angle $\alpha$ (which is $52^\circ$) form a linear pair (they are supplementary), so: $x + \alpha = 180^\circ$ Substitute $\alpha = 52^\circ$: $x + 52^\circ = 180^\circ$ $x = 180^\circ - 52^\circ = 128^\circ$

Thus, the value of $x$ is $\boxed{128^\circ}$.

Would you like more details or have any questions? Here are some related questions to explore further:

1. What are alternate interior angles, and why are they equal?
2. How do you determine if two angles are supplementary?
3. What are the properties of linear pairs in geometry?
4. Can you calculate the angle $\alpha + 26^\circ$ directly from the diagram?
5. What role do parallel lines play in the equality of angles?
6. How would the problem change if $\alpha + 26^\circ$ was not equal to $78^\circ$?
7. What is the relationship between the angles formed by a transversal cutting two parallel lines?
8. How can the sum of angles in a triangle be used to find missing angles?

Tip: Always look for relationships between angles such as supplementary, complementary, and vertical angles when solving problems involving intersecting lines and angles.