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Let's solve the problem shown in the image step-by-step.

### Problem 10
You have two parallel lines \( u \) and \( v \) cut by a transversal. The two angles on opposite sides of the transversal are:
- \( 7x + 2 \)
- \( 12x + 7 \)

Since lines \( u \) and \( v \) are parallel, these are corresponding angles, which means they are equal. Therefore, we set up the equation:
\[
7x + 2 = 12x + 7
\]

#### Step-by-Step Solution:
1. **Rearrange the equation to isolate terms with \( x \):**
   \[
   7x - 12x = 7 - 2
   \]
2. **Simplify both sides:**
   \[
   -5x = 5
   \]
3. **Solve for \( x \):**
   \[
   x = -1
   \]

So, the value of \( x \) is:
\[
x = -1
\]

### Problem 12
The question asks for the value of \( x \) that makes \( f \parallel g \), and to identify the theorem that justifies this.

When two lines are parallel and cut by a transversal, corresponding angles are equal. Based on the given problem, this should also involve corresponding angles or alternate interior/exterior angles depending on the specific placement of angles \( 7x + 2 \) and \( 12x + 7 \) in the context of lines \( f \) and \( g \).

Let's summarize possible relevant theorems:
- **Corresponding Angles Postulate**: If two parallel lines are cut by a transversal, then each pair of corresponding angles is equal.
- **Alternate Interior Angles Theorem**: If two parallel lines are cut by a transversal, then each pair of alternate interior angles is equal.

### Answer Summary:
For Problem 10, we found that:
\[
x = -1
\]

Would you like further details or have any questions about the process? Here are some related questions you might explore:

1. What is the Corresponding Angles Postulate?
2. How does the Alternate Interior Angles Theorem apply to parallel lines?
3. Can you provide an example of a proof using corresponding angles?
4. How are alternate exterior angles used in proving lines are parallel?
5. What other angle relationships exist when a transversal cuts parallel lines?

**Tip:** When working with parallel lines and transversals, remember that corresponding, alternate interior, and alternate exterior angles are equal.

For what value of x is f || g? Which theorem justifies your answer?

Learn how to solve for x using corresponding angles in a parallel lines problem. Explore key algebraic concepts and theorems like the Corresponding Angles Postulate and Converse. Suitable for Grades 8-10.

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