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This problem shows two intersecting lines, which form vertical angles. Vertical angles are always congruent, meaning they are equal in measure.

The given angles are:

- One angle is \( 81^\circ \).
- The other angle is \( (3 + 4x)^\circ \).

Since the angles are vertical, we can set the expressions equal to each other:

\[
81 = 3 + 4x
\]

### Solving for \( x \):

1. Subtract 3 from both sides:

\[
81 - 3 = 4x
\]
\[
78 = 4x
\]

2. Divide both sides by 4:

\[
x = \frac{78}{4} = 19.5
\]

Thus, the value of \( x \) is \( 19.5 \).

Would you like more details or have any other questions?

### Related questions:
1. How do you find vertical angles in any intersection of lines?
2. What is the definition of supplementary angles, and how do they relate to this problem?
3. If one angle was \( (2x + 5)^\circ \), how would you set up the equation for \( x \)?
4. How can you verify your solution in problems involving vertical angles?
5. What would happen if the problem involved supplementary instead of vertical angles?

### Tip:
Remember that vertical angles are always equal, which is a key property when solving equations involving intersecting lines!

Find the value of x given two vertical angles: one is 81° and the other is (3 + 4x)°.

Find the value of x by equating two vertical angles: 81° and (3 + 4x)°. This problem demonstrates key concepts from algebra and geometry, applying the Vertical Angles Theorem. Suitable for students in Grades 8-10.

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