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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.

Given m || n, find the value of x. The angles (3x - 23)° and (7x - 7)° are alternate interior angles formed by a transversal.

Find the value of x when given parallel lines m || n and a transversal. The angles formed are (3x - 23)° and (7x - 7)°. This problem explores the use of the Alternate Interior Angles Theorem and involves basic algebra to solve for x. Suitable for students in Grades 8-10.

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