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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.

Solve for x given that vertical angles are formed by two intersecting lines, where one angle is 63° and the other is (2x + 1)°.

In this problem, learn how to solve for x when two intersecting lines form vertical angles. The given angles are (2x + 1)° and 63°, and vertical angles are always equal. Follow a step-by-step algebraic solution. Suitable for students in Grades 7-9.

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