The image shows a diagram with two parallel lines (presumably $PV$ and $QT$) intersected by a transversal line $VW$. The problem asks to solve for $x$ given the angles:

• Angle $\angle PVW = (8x - 31)^\circ$
• Angle $\angle WQT = (5x + 17)^\circ$

These angles are corresponding angles, meaning they are congruent because $PV \parallel QT$ and $VW$ is a transversal.

Therefore, we can set up the following equation:

$8x - 31 = 5x + 17$

Solving for $x$:

1. Subtract $5x$ from both sides: $8x - 5x - 31 = 17$ $3x - 31 = 17$
2. Add 31 to both sides: $3x = 48$
3. Divide by 3: $x = 16$

Thus, $x = 16$.

Would you like further details on this? Here are 5 related questions to consider:

1. How do corresponding angles relate when a transversal intersects two parallel lines?
2. Can you explain why corresponding angles are congruent?
3. What are alternate interior angles, and how do they differ from corresponding angles?
4. How would the solution change if the angles were supplementary instead of congruent?
5. How can you check if two lines are parallel based on the angle relationships?

Tip: When dealing with parallel lines and a transversal, remember that corresponding, alternate interior, and consecutive interior angles can help determine angle relationships.