The image shows two parallel lines cut by a transversal. The angles formed are labeled as $x^\circ$ and $\left(\frac{1}{2}x + 15\right)^\circ$. Based on the positioning of these angles, they appear to be alternate interior angles, which are congruent when the lines are parallel.

This means that the two angles are equal:

$x = \frac{1}{2}x + 15$

Now, let's solve for $x$.

Step 1: Set up the equation

$x = \frac{1}{2}x + 15$

Step 2: Subtract $\frac{1}{2}x$ from both sides

$x - \frac{1}{2}x = 15$

Step 3: Simplify

$\frac{1}{2}x = 15$

Step 4: Multiply both sides by 2 to solve for $x$

$x = 30$

Thus, the value of $x$ is $30^\circ$.

Would you like further details or have any questions about this problem? Here are five related questions you might find interesting:

1. How can we identify corresponding angles when two parallel lines are cut by a transversal?
2. What other angle pairs are formed when parallel lines are intersected by a transversal?
3. Can we apply this approach if the lines were not parallel? Why or why not?
4. How would the solution change if the angles were complementary instead of equal?
5. What other geometric properties can we derive from parallel lines and transversals?

Tip: Always check for congruent or supplementary angle relationships when working with parallel lines cut by a transversal.