https://r2.erweima.ai/mathbot/image/1727029713097juyng3jx.jpg

The image shows a geometric diagram where two parallel lines are intersected by a transversal, and the task is to find the values of \( z \) and \( y \). 

Here’s a breakdown:

1. You are given that two lines are parallel: \( l \parallel n \).
2. The angles formed by the transversal and the parallel lines are labeled:
   - One of the angles is \( 66^\circ \),
   - Another angle is labeled \( z^\circ \),
   - A third angle is expressed as \( (3y + 51)^\circ \).

### Step 1: Use of Corresponding Angles
Since the lines are parallel, corresponding angles are equal. The angle labeled \( 66^\circ \) and the angle \( z^\circ \) are corresponding angles.

Thus:
\[
z = 66^\circ
\]

### Step 2: Solve for \( y \)
Next, we consider the angle \( z^\circ \) and the angle labeled \( (3y + 51)^\circ \). These two angles form a linear pair (they are supplementary because they lie on a straight line), so their sum must be \( 180^\circ \).

Thus, we have the equation:
\[
z + (3y + 51) = 180^\circ
\]
Substituting \( z = 66^\circ \) into the equation:
\[
66 + (3y + 51) = 180
\]
Simplifying:
\[
66 + 3y + 51 = 180
\]
\[
117 + 3y = 180
\]
\[
3y = 180 - 117
\]
\[
3y = 63
\]
\[
y = 21
\]

### Final Answer:
\[
z = 66^\circ, \quad y = 21
\]

Would you like more details, or do you have any further questions? 

Here are five related questions to deepen your understanding:
1. What are alternate interior angles, and how can they be identified in parallel line problems?
2. How can the concept of vertical angles help solve geometry problems with transversals?
3. Can you explain why corresponding angles are equal when lines are parallel?
4. What would change if the lines were not parallel?
5. How can you prove that the sum of angles on a straight line is always \(180^\circ\)?

**Tip**: When dealing with parallel lines and transversals, always look for angle relationships like corresponding, alternate interior, or supplementary angles.

En la figura a continuación, l || n. Hallar los valores de z y y.

Learn how to find the values of z and y using angle relationships in parallel lines intersected by a transversal. This geometry problem involves corresponding and supplementary angles, suitable for students in Grades 8-10.

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