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Given m, \parallel, n*m*∥*n*, find the value of x*x* and y, .*y*.

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.

Learn how to find the values of x and y in a parallel lines geometry problem using corresponding and supplementary angles. This detailed solution covers key geometry concepts suitable for Grades 8-10.

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