In this image, two parallel lines $\ell_1$ and $\ell_2$ are intersected by a transversal $t$, creating two angles: $(14x + 12)^\circ$ and $(8x + 14)^\circ$.

Step-by-Step Solution:

By the Corresponding Angles Postulate, angles formed by a transversal with two parallel lines are congruent (equal) if they are in corresponding positions. Therefore:

$(14x + 12)^\circ = (8x + 14)^\circ$

Solving for $x$:

1. Set up the equation:

$14x + 12 = 8x + 14$

2. Subtract $8x$ from both sides to simplify:

$14x - 8x + 12 = 14$ $6x + 12 = 14$

3. Subtract 12 from both sides:

$6x = 2$

4. Divide both sides by 6:

$x = \frac{2}{6} = \frac{1}{3}$

Thus, $x = \frac{1}{3}$.

Would you like more details on the properties of parallel lines or any other concepts in this problem?

Here are some related questions:

1. What are alternate interior angles, and how would they be used in a similar problem?
2. What happens if the lines $\ell_1$ and $\ell_2$ are not parallel?
3. How can you check your solution using substitution?
4. How does the Corresponding Angles Postulate relate to other angle relationships in parallel lines?
5. Can you solve this problem graphically instead of algebraically?

Tip: For problems involving parallel lines, always consider angle relationships like corresponding, alternate interior, and exterior angles—they can greatly simplify the problem.