In this problem, we are given that lines $m \parallel n$, meaning they are parallel, and we need to find the values of $x$ and $y$.

Step 1: Recognize angle relationships due to parallel lines

Since the lines $m$ and $n$ are parallel and a transversal cuts across them, the angles labeled $(8x + 2)^\circ$ and $(4x + 10)^\circ$ form corresponding angles. Corresponding angles are equal when the lines are parallel.

Thus, we can set up the equation: $(8x + 2)^\circ = (4x + 10)^\circ$

Step 2: Solve for $x$

We solve the equation: $8x + 2 = 4x + 10$ Subtract $4x$ from both sides: $4x + 2 = 10$ Subtract 2 from both sides: $4x = 8$ Divide both sides by 4: $x = 2$

Step 3: Solve for $y$

Since $y^\circ$ and $(4x + 10)^\circ$ are alternate interior angles (which are equal when the lines are parallel), we can write: $y = 4x + 10$ Substitute $x = 2$ into this equation: $y = 4(2) + 10 = 8 + 10 = 18$

Final Answer:

$x = 2, \quad y = 18$

Would you like any further explanation or details?

Related Questions:

1. What is the definition of corresponding angles?
2. How do you prove that alternate interior angles are equal when lines are parallel?
3. Can you explain the significance of transversal lines in geometry?
4. How does the value of $x$ affect the angle equations?
5. What would happen if the lines were not parallel?

Tip: When working with parallel lines, always look for corresponding or alternate interior angles, as these can be easily set equal to solve for unknown variables.