We are given that lines $s \parallel t$ and we need to find the value of $x$ given the angles.

The diagram shows a pair of parallel lines $s \parallel t$ cut by a transversal. Angles $\angle 3$ and $\angle 12$ are given in terms of $x$, and these two angles are corresponding angles. Since corresponding angles are congruent when lines are parallel, we have the equation:

$m\angle 3 = m\angle 12$

Substitute the expressions for the angles:

$6x + 12 = 10x + 24$

Now, solve for $x$:

1. Subtract $6x$ from both sides:

$12 = 4x + 24$

2. Subtract 24 from both sides:

$-12 = 4x$

3. Divide by 4:

$x = -3$

Thus, the value of $x$ is $\boxed{-3}$.

Would you like further details or have any questions?

Here are some related questions to expand on this:

1. What are the properties of corresponding angles when lines are parallel?
2. How would the equation change if angles 3 and 12 were supplementary instead of corresponding?
3. Can alternate interior angles also be used to solve this problem?
4. What other types of angle relationships exist when two lines are parallel and cut by a transversal?
5. How do you check if two lines are parallel using angles formed by a transversal?

Tip: Always remember that corresponding angles are equal when two lines are parallel and cut by a transversal.