This is a geometry problem involving parallel lines and transversals. The angles labeled with expressions must satisfy relationships like alternate interior angles, corresponding angles, or supplementary angles.

1. Relation between $27x + 4$ and $8x + 1$: These are alternate interior angles, so they are equal.
   $27x + 4 = 8x + 1$

2. Relation between $y + 10$ and $27x + 4$: These are supplementary angles because they form a straight line.
   $(27x + 4) + (y + 10) = 180$

Let's solve these equations step by step.

Step 1: Solve for $x$

$27x + 4 = 8x + 1$ $27x - 8x = 1 - 4$ $19x = -3$ $x = -\frac{3}{19}$

Step 2: Solve for $y$

Substitute $x = -\frac{3}{19}$ into the supplementary equation: $(27x + 4) + (y + 10) = 180$ $27\left(-\frac{3}{19}\right) + 4 + y + 10 = 180$ $-\frac{81}{19} + 4 + y + 10 = 180$ $y + \left(-\frac{81}{19} + 14\right) = 180$ Simplify: $-\frac{81}{19} + 14 = \frac{-81 + 266}{19} = \frac{185}{19}$ $y + \frac{185}{19} = 180$ $y = 180 - \frac{185}{19} = \frac{3420}{19} - \frac{185}{19} = \frac{3235}{19}$

Final Answer:

$x = -\frac{3}{19}, \quad y = \frac{3235}{19}$

Would you like a detailed verification or a step-by-step breakdown? Let me know!

Here are 5 related questions for practice:

1. How do alternate interior angles behave in parallel lines?
2. How are supplementary angles used to solve geometry problems?
3. Can you generalize how to solve for variables in angle relationships?
4. How would the solution change if the angle relationships were corresponding angles?
5. Why is it important to simplify fractions for accurate final answers?

Tip: Always double-check whether the given angles are equal or supplementary before solving equations.