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.