The problem appears to focus on determining whether certain lines in the diagram are parallel, likely by analyzing angles and relationships using geometry rules such as corresponding angles, alternate interior angles, or the sum of angles on a straight line.

Observations from the image:

1. Key Angles and Expressions:

   • One of the angles is marked as $x + 30^\circ$.
   • Another pair of angles is labeled $3x - 40^\circ$ and $2x + 20^\circ$.

2. Calculation for $x$: The handwritten solution indicates that the equation $3x + 30^\circ = 180^\circ$ was used to solve for $x$. This comes from the fact that the sum of angles on a straight line equals $180^\circ$.

   Solving: $3x + 30 = 180$ $3x = 150$ $x = 50^\circ$

3. Substitute $x = 50^\circ$:

   • $x + 30 = 50 + 30 = 80^\circ$,
   • $3x - 40 = 3(50) - 40 = 150 - 40 = 110^\circ$,
   • $2x + 20 = 2(50) + 20 = 100 + 20 = 120^\circ$.

4. Parallelism Check: The problem likely asks which lines (e.g., $a \parallel b$, $m \parallel n$, etc.) are parallel by analyzing the relationships between these angles.

Explanation of Relationships:

• For lines to be parallel, certain angle pairs must be equal:
  • Corresponding angles (e.g., if $\angle (x + 30) = \angle (3x - 40)$),
  • Alternate interior angles,
  • Supplementary angles for consecutive interior angles.

Based on the values calculated, a comparison of the angle pairs will confirm parallelism. Let me know if you’d like to proceed step-by-step through this reasoning.

Would you like a detailed solution for proving which lines are parallel?

5 Follow-up Questions:

1. What are the conditions for two lines to be parallel in terms of angles?
2. How does solving for $x$ help in determining the relationships between the angles?
3. Could the given solution be verified using another angle relationship?
4. What role do the alternate interior or corresponding angles play here?
5. How can you systematically compare all angle relationships to conclude parallelism?

Tip:

Always check the sums of angles on a straight line and at a point to confirm accuracy when working with parallel line problems!