To solve for $x$ in this problem, we need to recognize that lines $P$ and $Q$ are parallel and are intersected by transversal $R$. This setup creates pairs of corresponding angles that are congruent.

In this diagram:

• The angle labeled $(3x)^\circ$ on line $P$ and the angle labeled $\left( \frac{1}{2}x + 105 \right)^\circ$ on line $Q$ are corresponding angles. Since $P$ and $Q$ are parallel, these angles must be equal.

Thus, we can set up the following equation: $3x = \frac{1}{2}x + 105$

Solving for $x$:

1. Eliminate the fraction by multiplying every term by 2: $2 \cdot 3x = 2 \cdot \frac{1}{2}x + 2 \cdot 105$ which simplifies to: $6x = x + 210$

2. Isolate $x$ by subtracting $x$ from both sides: $6x - x = 210$ $5x = 210$

3. Solve for $x$ by dividing both sides by 5: $x = 42$

Conclusion:

The value of $x$ is $42$. Thus, the correct answer is:

• A. 42

Would you like further details or have any questions?

Here are five related questions for further practice:

1. How do corresponding angles relate when two parallel lines are cut by a transversal?
2. What are alternate interior angles, and how do they compare in parallel line setups?
3. How would the equation change if one of the angles were labeled differently?
4. Can we apply similar techniques to solve for angles when lines are not parallel?
5. How would you verify your answer by substituting back into the angles?

Tip: When dealing with parallel lines and a transversal, remember that corresponding, alternate interior, and alternate exterior angles are equal.