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In the given figure, we are asked to find the values of \( y \) and \( z \) with the condition that lines \( m \parallel l \) (meaning that lines \( m \) and \( l \) are parallel). We are also given the following angles:

- \( y^\circ \), which is an unknown angle.
- \( 64^\circ \), an adjacent angle.
- \( (6z - 10)^\circ \), an expression for another angle.

### Step 1: Identifying relationships between angles

1. **Corresponding angles**: Since \( m \parallel l \), and \( g \) is a transversal line, the angles on the opposite sides of the transversal at corresponding positions are equal.
   - This means \( y^\circ = 64^\circ \) because they are corresponding angles.

### Step 2: Solving for \( z \)

2. **Linear pair**: The angles \( (6z - 10)^\circ \) and \( y^\circ \) form a linear pair because they are on a straight line. Therefore, the sum of these two angles must equal 180°.
   - This gives the equation:
   \[
   (6z - 10) + y = 180
   \]
   Substituting \( y = 64 \), we have:
   \[
   (6z - 10) + 64 = 180
   \]
   Simplifying:
   \[
   6z - 10 + 64 = 180
   \]
   \[
   6z + 54 = 180
   \]
   \[
   6z = 180 - 54
   \]
   \[
   6z = 126
   \]
   \[
   z = \frac{126}{6} = 21
   \]

### Final Results:
- \( y = 64^\circ \)
- \( z = 21 \)

Would you like further details or have any questions? 

Here are five related questions to deepen your understanding:
1. How are corresponding angles defined for parallel lines and a transversal?
2. What is a linear pair of angles, and why must their sum be 180°?
3. How would the angle relationships change if the lines were not parallel?
4. Can alternate interior angles be used in this problem? Why or why not?
5. How can the value of \( z \) be used to check the consistency of the solution?

**Tip:** Always remember that parallel lines create multiple types of angle relationships (corresponding, alternate interior, etc.), which can be very useful in solving for unknown angles.

En la figura siguiente, m || l. Hallar los valores de y y z.

Learn how to solve for the angles y and z using the corresponding angles theorem and the linear pair theorem. The problem involves parallel lines and a transversal, with step-by-step reasoning for students in Grades 7-9.

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