To solve for $x$ and $y$ given that $m \parallel n$, we can use properties of parallel lines and transversals, specifically that:

1. Corresponding angles are equal.
2. Alternate interior angles are equal.

From the diagram:

• Since $m \parallel n$, the angles on corresponding positions are equal.

Step 1: Set up equations based on corresponding angles.

1. The angle labeled $(3x - 20)^\circ$ corresponds with the angle labeled $(2x + 3)^\circ$.

   So, we set up the equation: $3x - 20 = 2x + 3$

2. The angle labeled $(y - 6)^\circ$ corresponds with the angle labeled $(2x + 3)^\circ$.

   So, we set up the equation: $y - 6 = 2x + 3$

Step 2: Solve for $x$ from the first equation.

1. Starting with: $3x - 20 = 2x + 3$

2. Subtract $2x$ from both sides: $x - 20 = 3$

3. Add 20 to both sides: $x = 23$

Step 3: Substitute $x = 23$ into the second equation to solve for $y$.

1. Substitute $x = 23$ into: $y - 6 = 2x + 3$

2. Calculate $2x + 3$ when $x = 23$: $2(23) + 3 = 46 + 3 = 49$

3. Substitute this back: $y - 6 = 49$

4. Add 6 to both sides: $y = 55$

Final Answer:

The values of $x$ and $y$ are: $x = 23, \quad y = 55$

Would you like more details or have any questions? Here are 5 related questions:

1. How can we check our solution by substituting back into the angle measures?
2. What properties of parallel lines allow us to solve for $x$ and $y$ here?
3. How would the solution change if the lines were not parallel?
4. Can you use the same approach to solve if different angles were given?
5. How would you find the sum of all angles in the intersection?

Tip: When working with parallel lines, corresponding and alternate interior angles are always key!