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To solve for \( x \) so that line \( A \parallel B \), we need to use the fact that the angles formed by a transversal cutting two parallel lines must be congruent (i.e., equal).

The two expressions for the angles in the diagram are given as:

- Angle on line \( A \): \( 4x + 14 \)
- Angle on line \( B \): \( 3x + 21 \)

Since the lines \( A \) and \( B \) must be parallel, these two angles must be equal.

### Step-by-step solution:

Set the expressions for the angles equal to each other:
\[
4x + 14 = 3x + 21
\]

Now, solve for \( x \):

1. Subtract \( 3x \) from both sides:
\[
4x - 3x + 14 = 21
\]
\[
x + 14 = 21
\]

2. Subtract 14 from both sides:
\[
x = 21 - 14
\]
\[
x = 7
\]

### Final answer:
\[
x = 7
\]

Let me know if you'd like further details!

Here are some related questions:
1. What would happen if the angles were supplementary instead of equal?
2. How do you determine when lines are perpendicular using angles?
3. What are the properties of alternate interior angles in parallel lines?
4. How do you solve when three parallel lines are intersected by a transversal?
5. What are the conditions for two lines to be parallel based on slopes?

**Tip**: Always double-check your work by substituting your solution back into the original equation.

Solve for x to make A || B. Given angles formed by a transversal: 4x + 14 (on line A) and 3x + 21 (on line B).

Learn how to solve for x to make two lines parallel by using the fact that corresponding angles formed by a transversal are equal. This problem involves basic algebraic manipulation and is suitable for students in Grades 8-10.

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