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In the given diagram, \( l \) and \( m \) are parallel lines, and we are asked to find the value of angle \( x \).

We can use the fact that corresponding and alternate interior angles are equal when two lines are parallel, and also that the sum of angles on a straight line is 180°.

Here’s the solution:

1. The angle adjacent to the 54° angle on the line \( l \) (where the straight angle forms) is \( 180^\circ - 54^\circ = 126^\circ \), since angles on a straight line sum to 180°.

2. Since \( l \parallel m \), the 126° angle is corresponding to the angle next to the 20° angle, meaning they are equal. Therefore, the angle adjacent to the 20° angle on line \( m \) is also 126°.

3. Finally, the interior angles of the triangle must sum to 180°. The triangle formed by the angles \( 126^\circ \), \( 20^\circ \), and \( x \) gives us the equation:
   \[
   126^\circ + 20^\circ + x = 180^\circ
   \]
   Simplifying:
   \[
   146^\circ + x = 180^\circ
   \]
   \[
   x = 180^\circ - 146^\circ = 34^\circ
   \]

Thus, the value of \( x \) is \( 34^\circ \).

Would you like any further details or clarifications?

Here are 5 related questions:
1. What are alternate interior angles, and why are they equal in parallel lines?
2. How do corresponding angles work in parallel line problems?
3. Can this problem be solved using the concept of exterior angles of a triangle?
4. What other properties of parallel lines are useful in geometry?
5. How can this concept be applied in real-world engineering problems?

**Tip**: When dealing with parallel lines, always look for corresponding or alternate interior angles to simplify your work.

Find the value of angle x when lines l and m are parallel, and you are given two angles 54° and 20° in the diagram.

Solve for angle x using the properties of parallel lines, corresponding angles, and the triangle sum theorem. This problem involves angles 54°, 20°, and parallel lines to find the missing angle in a triangle. Suitable for students in Grades 8-10.

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