Since $\angle 1$ and $\angle 2$ are vertical angles, they are equal. Therefore, we can set their measures equal to each other and solve for $x$.

Given: $m\angle 1 = (6x + 29)^\circ$ $m\angle 2 = (7x + 25)^\circ$

Since $\angle 1 = \angle 2$, we can write the equation:

$6x + 29 = 7x + 25$

Now, let's solve for $x$:

1. Subtract $6x$ from both sides: $29 = x + 25$

2. Subtract 25 from both sides: $4 = x$

So, the value of $x$ is:

$\boxed{4}$

Would you like more details or have any other questions?

Related Questions:

1. What are vertical angles, and why are they always equal?
2. How can we check if our solution for $x$ is correct?
3. What happens if vertical angles are not equal in a given problem?
4. How do we solve for angles when given expressions for their measures?
5. How can we find the measures of the angles after solving for $x$?

Tip: Vertical angles are always equal, making them useful in solving equations involving unknowns in geometry problems!