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The problem involves finding the value of \( x \) given two expressions for angles that are vertically opposite. Here is the provided information:

1. \( m \angle 1 = 4x + 2 \)
2. \( m \angle 3 = 5x - 15 \)

### Solution:

Since angles 1 and 3 are vertically opposite angles, they are equal. Therefore, we can set up the following equation:

\[
4x + 2 = 5x - 15
\]

Let's solve for \( x \):

1. Subtract \( 4x \) from both sides:

\[
2 = x - 15
\]

2. Add 15 to both sides:

\[
17 = x
\]

So, the value of \( x \) is **17**.

### Answer:
The correct answer is **D. 17**.

Would you like more details, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. What are vertically opposite angles, and why are they always equal?
2. How do you solve linear equations with variables on both sides?
3. Can you provide another example of a problem involving vertically opposite angles?
4. What other types of angles can be found when two lines intersect?
5. How can understanding angle relationships help in geometry problems?

**Tip:** Remember, vertically opposite angles are always equal when two lines intersect. This property is very useful in solving problems involving intersecting lines.

If m∠1 = (4x + 2) and m∠3 = (5x - 15), what is the value of x?

Find the value of x by applying the Vertically Opposite Angles Theorem in this geometry problem. Learn to solve linear equations with variables on both sides. Suitable for students in Grades 7-9.

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