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The diagram shows several lines and angles. Given that \( m_1 \parallel m_2 \) and \( m_3 \parallel m_4 \), we can apply properties of parallel lines and the angles they create with transversals to find \( x \).

1. **\( m_1 \parallel m_2 \) Property**:
   - The given \( 42^\circ \) angle formed by \( m_2 \) and the transversal is corresponding to the angle between \( m_1 \) and the transversal on the right, hence that angle is also \( 42^\circ \).

2. **\( m_3 \parallel m_4 \) Property**:
   - Since \( m_3 \parallel m_4 \), and using the fact that \( m_2 \) acts as a transversal, the angle \( x \) is alternate interior with the \( 42^\circ \) angle at \( m_1 \) because they are formed between parallel lines \( m_1 \) and \( m_3 \) with \( m_2 \) as a transversal.

Given that \( x \) is alternate interior with \( 42^\circ \), it must also be \( 42^\circ \).

Thus, the value of \( x \) is \( 42^\circ \).

Would you like further details on any of these concepts or have any questions about this solution? Here are a few related questions you might consider:

1. How do corresponding angles work when lines are parallel?
2. What are alternate interior angles?
3. Can you explain how the angle of \( 138^\circ \) and \( 158^\circ \) might be used or calculated in different geometric setups?
4. How would the angle measures change if the lines were not parallel?
5. What is the significance of transversal lines in geometry?

**Tip**: In problems involving parallel lines, always consider the relationship of corresponding, alternate interior, and consecutive interior angles, as these often provide the quickest path to a solution.

In the figure below, m1 || m2 and m3 || m4. What is the value of x?

Learn how to find the value of an unknown angle formed by parallel lines and a transversal. This problem involves key geometric concepts such as corresponding and alternate interior angles, suitable for students in Grades 8-10.

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