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In the given diagram, we are told that lines \( m \parallel n \) (i.e., they are parallel), and we are tasked with finding the value of \( x \).

From the diagram:
- The angle on the top is \( (5x + 3)^\circ \),
- The angle on the bottom is \( (x + 3)^\circ \).

Since the lines are parallel, and the angles are positioned as alternate interior angles (formed by a transversal intersecting the parallel lines), these angles must be congruent. Therefore, we can set the two angles equal to each other:

\[
5x + 3 = x + 3
\]

Now, let's solve for \( x \):

1. Subtract \( x \) from both sides:
\[
5x - x + 3 = 3
\]
\[
4x + 3 = 3
\]

2. Subtract 3 from both sides:
\[
4x = 0
\]

3. Divide both sides by 4:
\[
x = 0
\]

Thus, the value of \( x \) is \( 0 \).

Would you like more details or have any questions? 

Here are 5 related questions for further practice:
1. What would the value of \( x \) be if the angles were supplementary instead of congruent?
2. Can you identify other angle relationships in parallel lines cut by a transversal?
3. How would the equation change if one angle were \( (5x + 3)^\circ \) and the other were \( (2x + 15)^\circ \)?
4. What is the sum of the interior angles formed by a transversal cutting parallel lines?
5. How can you prove that alternate interior angles are always congruent for parallel lines?

**Tip**: Always look for congruent or supplementary angles when dealing with parallel lines cut by a transversal.

Given m parallel to n, find the value of x.

This problem involves finding the value of x using the Alternate Interior Angles Theorem. Given two parallel lines cut by a transversal, the angles (5x + 3)° and (x + 3)° are congruent, leading to an equation that can be solved step-by-step. Suitable for Grades 8-10 geometry students.

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