The image shows two intersecting lines forming vertical angles. One angle is given as $63^\circ$, and the other is expressed algebraically as $(2x + 1)^\circ$. To find $x$, we can use the property of vertical angles, which states that vertical angles are equal.

Step-by-Step Solution:

1. Identify the relationship: Vertical angles are equal. Therefore, we set the two angles equal to each other: $(2x + 1)^\circ = 63^\circ$

2. Solve for $x$: $2x + 1 = 63$

   Subtract 1 from both sides: $2x = 62$

   Divide by 2: $x = 31$

Answer:

$x = 31$.

Would you like any further details or explanations?

Related Questions:

1. What are vertical angles, and why are they equal?
2. How would the equation change if the angles were supplementary?
3. Can vertical angles ever be complementary?
4. What other geometric properties apply when two lines intersect?
5. How would the solution change if the equation were $(2x + 1)^\circ = 90^\circ$?

Tip: Always look for angle relationships such as vertical angles, complementary angles, and supplementary angles when dealing with intersecting lines.