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To solve for \( x \) in this problem, we can observe that the angles given, \( (3x + 39)^\circ \) and \( (8x - 16)^\circ \), are vertically opposite angles. Vertically opposite angles are always equal, so we set them equal to each other:

\[
3x + 39 = 8x - 16
\]

### Step-by-Step Solution:

1. **Move all terms involving \( x \) to one side:**
   \[
   39 + 16 = 8x - 3x
   \]
   \[
   55 = 5x
   \]

2. **Divide by 5 to isolate \( x \):**
   \[
   x = \frac{55}{5}
   \]
   \[
   x = 11
   \]

### Conclusion:
The value of \( x \) is \( 11 \).

Would you like more details on any step, or have any questions?

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Here are 5 related questions to expand understanding:
1. What are vertically opposite angles and why are they always equal?
2. If one angle was instead labeled \( (5x + 10)^\circ \), how would that change the solution?
3. Can you find the actual angle measurements when \( x = 11 \)?
4. What would happen if the angles were not vertically opposite but adjacent?
5. How do we solve for \( x \) if we were given complementary or supplementary angles?

**Tip:** Vertically opposite angles are equal because they are formed by two intersecting lines, which creates two pairs of opposite angles with the same measurement.

Find the value of x, given that (3x + 39)° and (8x - 16)° are vertically opposite angles.

Learn how to find the value of x when given vertically opposite angles (3x + 39)° and (8x - 16)°. This step-by-step solution demonstrates key algebraic concepts and the Vertically Opposite Angles Theorem. Suitable for Grade 8 students.

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