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In this problem, we are given two intersecting lines, forming vertical angles. These two angles are described by expressions involving \( n \):

1. One angle is \( n + 116^\circ \),
2. The other angle is \( -4n + 126^\circ \).

Since vertical angles are congruent, we can set these two expressions equal to each other:

\[
n + 116 = -4n + 126
\]

Now, let's solve for \( n \):

### Step 1: Combine like terms.
Move all terms involving \( n \) to one side of the equation and constants to the other side:

\[
n + 4n = 126 - 116
\]

This simplifies to:

\[
5n = 10
\]

### Step 2: Solve for \( n \).
Divide both sides by 5:

\[
n = 2
\]

Thus, the value of \( n \) is \( \boxed{2} \).

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Would you like more details or have any questions?

Here are 5 questions that expand on this topic:
1. What are vertical angles, and why are they congruent?
2. How can you verify the solution using the sum of angles around a point?
3. What are alternate interior angles, and how are they related to this type of problem?
4. How would the equation change if one of the angles was an exterior angle?
5. Could this problem be solved using a different method, such as the sum of linear pairs?

**Tip:** Remember, vertical angles are always congruent, so whenever you encounter intersecting lines, equating the angles will help solve for unknowns.

Find the value of n given the angles (n + 116)° and (-4n + 126)° formed by intersecting lines.

Learn how to solve for n by applying the vertical angles theorem to the intersecting lines. This problem involves algebraic manipulation and angle properties, suitable for Grades 7-9.

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