https://file.aiquickdraw.com/mathbot/file/17266814808062ggz6uxm.jpg

The image shows two parallel lines \( m \parallel n \) cut by a transversal. This creates two angles: \( (x - 3)^\circ \) and \( (4x + 3)^\circ \). 

Since the lines \( m \) and \( n \) are parallel, the angles formed are **corresponding angles** or **alternate interior angles**, meaning they are equal.

Thus, we can set up the equation:
\[
(x - 3)^\circ = (4x + 3)^\circ
\]

Now, solving this equation:

\[
x - 3 = 4x + 3
\]

Subtract \( x \) from both sides:

\[
-3 = 3x + 3
\]

Subtract 3 from both sides:

\[
-6 = 3x
\]

Now, divide both sides by 3:

\[
x = -2
\]

The value of \( x \) is \( \boxed{-2} \).

Would you like any additional details or have any other questions? Here are 5 related questions to expand on this:

1. What are corresponding angles and how are they used in parallel line theorems?
2. Can you explain the concept of alternate interior angles?
3. What would change if the angles were on the same side of the transversal?
4. How can you verify that the lines are parallel using angle relationships?
5. How do you solve more complex equations involving angles in geometry?

**Tip:** Always check if lines are parallel when dealing with angles formed by a transversal, as this simplifies the problem significantly.

Explore the solution for x in parallel lines cut by a transversal using the corresponding angles postulate. This detailed example walks you through setting up and solving the equation (x - 3) = (4x + 3) to find x when lines are parallel.

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