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In this figure, the lines \( L_1 \) and \( L_2 \) are crossed by a transversal, creating angles that are related to each other. Divya wants to find the value of \( x \) using a geometric theorem.

The angles given are \( (x + 1)^\circ \) and \( (3x - 2)^\circ \). These two angles are **alternate interior angles** because they lie on opposite sides of the transversal and inside the parallel lines \( L_1 \) and \( L_2 \). According to the **Alternate Interior Angles Theorem**, when two parallel lines are cut by a transversal, the alternate interior angles are congruent (equal).

Therefore, the theorem that Divya can use to solve for \( x \) is:

**B. Alternate Interior Angles Theorem.**

Now, to solve for \( x \):

\[
x + 1 = 3x - 2
\]

Simplifying the equation:

\[
1 + 2 = 3x - x
\]

\[
3 = 2x
\]

\[
x = \frac{3}{2} = 1.5
\]

Thus, \( x = 1.5 \).

Would you like any further clarification or details?

Here are some related questions to expand on the concept:

1. How does the Alternate Interior Angles Theorem apply in other types of geometric problems?
2. Can the Alternate Exterior Angles Theorem be used in this scenario, and why or why not?
3. What happens if the lines are not parallel in such a problem?
4. How does the Triangle Sum Theorem differ from angle theorems involving parallel lines?
5. Can you provide an example using the Vertical Angles Theorem?

**Tip:** Always ensure that the lines are parallel when applying angle theorems related to parallel lines and a transversal.

What theorem can Divya use to solve for x?

Learn how to apply the Alternate Interior Angles Theorem to solve for x in a geometric problem involving parallel lines cut by a transversal. This problem covers key concepts in geometry and is suitable for students in Grades 8-10.

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