To solve for $x$, we start by noting that lines $m$ and $n$ are parallel, and the angles $(8x + 12)^\circ$ and $(10x + 6)^\circ$ are corresponding angles.

For corresponding angles, we know that: $8x + 12 = 10x + 6$

Step 1: Set up the equation

$8x + 12 = 10x + 6$

Step 2: Solve for $x$

1. Subtract $8x$ from both sides: $12 = 2x + 6$
2. Subtract 6 from both sides: $6 = 2x$
3. Divide both sides by 2: $x = 3$

Solution

The value of $x$ is $\mathbf{3}$.

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

Related Questions

1. What are corresponding angles, and why are they equal here?
2. How would the equation change if the angles were supplementary?
3. What other angle pairs are equal when two lines are parallel and cut by a transversal?
4. How would you check the solution to ensure $x = 3$ is correct?
5. How do alternate interior angles relate to parallel lines?

Tip: When solving angle equations, remember to check whether the angles should be equal or supplementary based on their relative positions.