In the given diagram, lines $m$ and $n$ are parallel, and the angles $(3x - 23)^\circ$ and $(7x - 7)^\circ$ are formed by a transversal crossing the parallel lines. These angles are alternate interior angles, which are congruent (equal) because $m \parallel n$.

Thus, we can set the two angles equal to each other and solve for $x$:

$3x - 23 = 7x - 7$

Now, solving for $x$:

$3x - 23 = 7x - 7$ $3x - 7x = -7 + 23$ $-4x = 16$ $x = \frac{16}{-4} = -4$

Thus, the value of $x$ is $x = -4$.

Would you like further explanation on any part of this solution?

Related Questions:

1. What are alternate interior angles, and why are they equal?
2. How does the concept of parallel lines influence the behavior of angles formed by a transversal?
3. Can you find the specific measures of each angle once you know $x = -4$?
4. What other angle relationships occur with parallel lines and transversals (e.g., corresponding angles)?
5. How would the problem change if the angles were not alternate interior angles?

Tip:

When working with parallel lines and transversals, identifying angle relationships (corresponding, alternate interior, or consecutive interior) helps in setting up equations.