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The image shows two parallel lines cut by a transversal. One of the angles is given as \( 110^\circ \), and we need to find the measure of the angle \( x^\circ \).

### Step-by-Step Solution:

1. **Identify the Relationship**:
   - The angle \( x^\circ \) and the angle \( 110^\circ \) are **alternate interior angles** because they are on opposite sides of the transversal and inside the parallel lines.
   
2. **Apply the Alternate Interior Angles Theorem**:
   - Alternate interior angles are congruent when two parallel lines are cut by a transversal.
   - Therefore, \( x^\circ = 110^\circ \).

### Final Answer:
\[
x = 110^\circ
\]

Would you like to proceed with solving more angles from the puzzle, or do you have any questions? Here are five related questions to expand your understanding:

1. What is the relationship between alternate exterior angles?
2. How would you find the value of \( x \) if the given angle was instead an exterior angle?
3. What are corresponding angles, and how are they related when parallel lines are involved?
4. How do consecutive interior angles relate to each other?
5. What would the value of \( x \) be if the given angle was \( 70^\circ \)?

**Tip:** Always remember that the sum of angles on a straight line is \( 180^\circ \), which is useful when dealing with linear pairs.

Explore how to solve angles in geometry using the Alternate Interior Angles Theorem. This problem involves two parallel lines cut by a transversal with one angle given as 110 degrees, and finding the measure of another angle x degrees. Suitable for students in Grades 7-9.

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