To solve the given problem where lines $m \parallel n$ and the angles are labeled, we need to apply the properties of parallel lines and corresponding angles.

Given:

• $m \parallel n$
• One angle is labeled $(6x + 4)^\circ$
• Another angle is labeled $(9x - 14)^\circ$
• The angle corresponding to the $y^\circ$ is the vertical angle of $(9x - 14)^\circ$.

Solution Approach:

1. Corresponding Angles: Since $m \parallel n$, the angles formed by the transversal are corresponding angles. Hence, $(6x + 4)^\circ = y^\circ$ because they are corresponding angles.

2. Supplementary Angles: The angles $(9x - 14)^\circ$ and $(6x + 4)^\circ$ are on a straight line (i.e., they are supplementary). Therefore, their sum must be $180^\circ$. This gives the equation: $(9x - 14) + (6x + 4) = 180$ Simplifying the equation: $9x - 14 + 6x + 4 = 180$ $15x - 10 = 180$ $15x = 190$ $x = \frac{190}{15} = 12.67$

3. Find $y$: Now that we have $x = 12.67$, substitute this into the expression for $y^\circ = (6x + 4)^\circ$: $y = 6(12.67) + 4 = 76 + 4 = 80^\circ$

Final Answer:

• $x = 12.67$
• $y = 80^\circ$

Let me know if you'd like any further clarification!

Here are five related questions you might explore:

1. What are alternate interior angles in a transversal with parallel lines?
2. How do corresponding angles relate in parallel lines cut by a transversal?
3. What is the significance of supplementary angles in geometry?
4. How can vertical angles help simplify geometric calculations?
5. How would this problem change if the lines were not parallel?

Tip: Always verify the relationship between angles in parallel line problems (e.g., corresponding, alternate interior, or supplementary) for a quicker solution.