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Problem Breakdown:

• Parallel Lines: $k$ and $m$ are parallel.
• Transversal Lines: $p$ and $q$ intersect the parallel lines.
• Angles Given:
  • One angle is $40^\circ$.
  • Another angle is $80^\circ$.
  • You are tasked with finding $x$, the unknown angle.

Approach:

1. Use Angle Relationships:

   • Alternate Interior Angles: When a transversal crosses parallel lines, alternate interior angles are equal.
   • Corresponding Angles: These are equal as well.
   • Supplementary Angles: On a straight line, the sum of adjacent angles equals $180^\circ$.

2. Solve for $x$:

   • Identify which angle relationships are relevant based on the diagram.

Let me calculate the solution based on these principles.# Calculation for solving the problem

Given angles

angle_1 = 40 # Given angle on one side of the transversal angle_2 = 80 # Adjacent angle on the same line as x

Sum of angles on a straight line

straight_line_sum = 180

Calculate x using the supplementary angle relationship

x = straight_line_sum - angle_2

x100100successThe value of $x$ is $100^\circ$.

Explanation:

1. The angle $80^\circ$ and $x$ lie on the same straight line, making them supplementary.
2. Using the supplementary angle relationship: $80^\circ + x = 180^\circ$.
3. Solving for $x$: $x = 180^\circ - 80^\circ = 100^\circ$.

Would you like further clarification, or do you want to explore related questions? Here are five follow-ups:

1. What are alternate interior and exterior angles, and how are they used in such problems?
2. How do you identify corresponding angles in parallel line diagrams?
3. Can you explain how to verify if two lines are parallel using angle relationships?
4. What happens if the lines are not parallel but intersect?
5. Can you provide similar examples with different given angles?

Tip: Always label all angles clearly when working with parallel lines and transversals to avoid confusion.