In this problem, we're given two parallel lines $m$ and $n$ and a transversal line that intersects them, creating angles labeled as $(5x + 2)^\circ$ and $(x - 2)^\circ$. Since $m \parallel n$, we can use properties of parallel lines and transversals to set up an equation.

Solution

The angles $(5x + 2)^\circ$ and $(x - 2)^\circ$ are alternate interior angles because they lie on opposite sides of the transversal and between the parallel lines. When two lines are parallel, alternate interior angles are equal.

So, we can set up the equation: $5x + 2 = x - 2$

Step-by-Step Solution

1. Subtract $x$ from both sides: $4x + 2 = -2$

2. Subtract 2 from both sides: $4x = -4$

3. Divide by 4: $x = -1$

Final Answer

The value of $x$ is $-1$.

Would you like a detailed explanation of why alternate interior angles are equal when lines are parallel, or do you have any questions?

Related Questions

1. What are alternate interior angles, and why are they equal in parallel lines?
2. If the angles were corresponding angles, how would the setup change?
3. How would you solve this if the given angles were supplementary instead of equal?
4. Can the value of $x$ ever be positive in such configurations? Why or why not?
5. What are other types of angle relationships formed by a transversal?

Tip

Remember that alternate interior angles are equal only when the lines are parallel. This is a key property used frequently in geometry problems involving transversals.