In this problem, the two lines $m \parallel n$ are parallel, and the angles formed by the transversal are given as:

• $(6x - 27)^\circ$
• $(8x - 3)^\circ$

Since the lines $m \parallel n$ are parallel, these angles are alternate interior angles, which are always equal. Therefore, we can set up the equation:

$6x - 27 = 8x - 3$

Solving for $x$:

1. Subtract $6x$ from both sides: $-27 = 2x - 3$

2. Add 3 to both sides: $-24 = 2x$

3. Divide both sides by 2: $x = -12$

Thus, the value of $x$ is $-12$.

Would you like more details on how these angles work in parallel lines, or do you have any other questions?

Here are 5 related questions to further explore this topic:

1. What are corresponding angles, and how do they differ from alternate interior angles?
2. How would you find $x$ if the angles given were exterior instead of interior angles?
3. What is the significance of the transversal line in geometry?
4. Can alternate exterior angles also be equal when lines are parallel?
5. What if the angles were supplementary instead of equal, how would the equation change?

Tip: Always remember that alternate interior angles are equal when two lines are parallel and intersected by a transversal.