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In the diagram, we have two parallel lines \( l \parallel m \) and a transversal intersecting them. The angles formed are \( (8x + 20)^\circ \) and \( (11x - 31)^\circ \), which are alternate interior angles.

Since \( l \parallel m \), alternate interior angles are congruent, meaning:

\[
8x + 20 = 11x - 31
\]

Now, solve for \( x \):

1. Subtract \( 8x \) from both sides:
\[
20 = 3x - 31
\]

2. Add 31 to both sides:
\[
51 = 3x
\]

3. Divide by 3:
\[
x = 17
\]

Thus, the value of \( x \) is \( \boxed{17} \).

Would you like more details on this? Here are some related questions:

1. What are alternate interior angles, and why are they congruent when lines are parallel?
2. How would the equation change if the angles were not alternate interior angles?
3. Can you verify the solution by substituting \( x = 17 \) back into the expressions for the angles?
4. What is the definition of a transversal in geometry?
5. How do we prove that two lines are parallel using angle relationships?

**Tip:** When working with parallel lines, remember that alternate interior angles, corresponding angles, and consecutive interior angles have specific relationships that can help solve for unknowns.

If l || m, find the value of x given the angles (8x + 20)° and (11x - 31)° formed by a transversal.

In this geometry problem, you will learn how to find the value of x by applying the Alternate Interior Angles Theorem. The angles formed by a transversal intersecting two parallel lines are (8x + 20)° and (11x - 31)°. Follow a step-by-step solution to solve for x, ideal for students in Grades 8-10.

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