To solve for $x$ and $y$, we first use the fact that $m \parallel n$. In parallel lines cut by a transversal, corresponding, alternate interior, or supplementary angles are related.

Observations:

1. Angle Relationships:

   • The angle $6x - 16$° is a vertical angle to $3x + 17$°, so they are equal:
     $6x - 16 = 3x + 17$

   • The angle $2y + 14$° and $6x - 16$° are consecutive interior angles (or supplementary), so: $(6x - 16) + (2y + 14) = 180$

Step 1: Solve for $x$

From the vertical angle equation: $6x - 16 = 3x + 17$ Simplify: $6x - 3x = 17 + 16$ $3x = 33$ $x = 11$

Step 2: Solve for $y$

Substitute $x = 11$ into the supplementary angle equation: $(6x - 16) + (2y + 14) = 180$ Substitute $6x - 16$: $(6(11) - 16) + (2y + 14) = 180$ $(66 - 16) + (2y + 14) = 180$ $50 + 2y + 14 = 180$ $2y + 64 = 180$ $2y = 116$ $y = 58$

Final Solution:

$x = 11, \quad y = 58$

Would you like a detailed explanation for any step or additional examples?

Related Questions:

1. What are the properties of angles formed by parallel lines and a transversal?
2. How can vertical angles be used to solve equations?
3. What happens if two consecutive interior angles are not supplementary?
4. How do we prove lines are parallel using angles?
5. Can this method work if the angle measures were in radians?

Tip:

Always check relationships between angles (e.g., supplementary, complementary, equal) to guide your equations!