https://file.aiquickdraw.com/mathbot/file/17300726576963ckh91ux.jpg

The image shows two intersecting lines, creating a pair of vertical (opposite) angles. The given angles are labeled as \((8w + 8)^\circ\) and \((9w - 5)^\circ\). Since they are vertical angles, they are congruent, meaning their measures are equal. 

Let's solve for \( w \):

\[
8w + 8 = 9w - 5
\]

### Step 1: Move all terms involving \( w \) to one side.
\[
8w - 9w = -5 - 8
\]
\[
-w = -13
\]

### Step 2: Divide by \(-1\) to isolate \( w \).
\[
w = 13
\]

So, the value of \( w \) is \( 13 \).

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

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Here are 5 related questions to explore further:

1. What are vertical angles, and why are they always equal?
2. How would we solve for \( w \) if the angle expressions had additional constants?
3. Can we determine the actual angle measure if \( w = 13 \)?
4. How do we know that the lines intersect to create vertical angles in this scenario?
5. What is the relationship between supplementary and vertical angles?

**Tip:** Always check for congruent angles when dealing with intersecting lines, as they often simplify the problem-solving process.

Solve for the value of w given the angle expressions (8w + 8)° and (9w - 5)° for intersecting lines.

Learn how to solve for the variable w in the context of vertical angles with expressions (8w + 8)° and (9w - 5)°. This geometry problem illustrates the application of the Vertical Angles Theorem and algebraic techniques, suitable for students in Grades 8-10.

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