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The image shows two parallel lines \( m \) and \( n \), cut by a transversal line \( t \). We are asked to find the value of angle \( x^\circ \), given that another angle on the opposite side of the transversal is \( 58^\circ \).

Since \( m \parallel n \) and \( t \) is a transversal, the angle \( x^\circ \) and the \( 58^\circ \) angle are alternate interior angles. By the Alternate Interior Angle Theorem, alternate interior angles are congruent when two parallel lines are cut by a transversal. Therefore, we can conclude:

\[
x^\circ = 58^\circ
\]

Thus, the value of \( x \) is \( 58^\circ \).

Would you like more details, or do you have any further questions?

Here are 5 related questions:
1. What is the Alternate Interior Angle Theorem?
2. How can you identify parallel lines and a transversal in geometry?
3. What are corresponding angles, and how do they relate to parallel lines?
4. How do you prove that two lines are parallel using angles?
5. Can alternate exterior angles be used in a similar situation?

**Tip:** When two lines are parallel and a transversal cuts them, several pairs of angles (like alternate interior, corresponding, and consecutive interior) have specific relationships. Understanding these can help solve for unknown angles efficiently.

Find the value of x when two parallel lines m and n are cut by a transversal, and one of the angles on the transversal is 58°.

Learn how to find the value of x when two parallel lines are cut by a transversal, given that one of the alternate interior angles is 58°. This problem applies the Alternate Interior Angle Theorem and is suitable for students in Grades 7-10.

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