https://file.aiquickdraw.com/mathbot/file/1726842301701h4xv3wgw.jpg

The image shows a pair of vertical angles, labeled \((x + 17)^\circ\) and \((4x - 34)^\circ\). Since vertical angles are congruent (they have the same measure), we can set these two expressions equal to each other and solve for \(x\).

### Step 1: Set the angles equal to each other
\[
x + 17 = 4x - 34
\]

### Step 2: Solve for \(x\)
- Subtract \(x\) from both sides:
\[
17 = 3x - 34
\]
- Add 34 to both sides:
\[
51 = 3x
\]
- Divide both sides by 3:
\[
x = 17
\]

### Step 3: Find the measure of each labeled angle
Now that we know \(x = 17\), substitute this value back into each angle expression:
- For \((x + 17)^\circ\):
\[
17 + 17 = 34^\circ
\]
- For \((4x - 34)^\circ\):
\[
4(17) - 34 = 68 - 34 = 34^\circ
\]

Thus, both angles measure \(34^\circ\).

### Final Answer:
- \(x = 17\)
- Each angle measures \(34^\circ\)

Would you like any further details or explanations?

Here are five related questions you might find interesting:
1. How do we know vertical angles are always congruent?
2. Can vertical angles ever be supplementary?
3. How would the solution change if the angles were supplementary instead of congruent?
4. What are alternate interior angles, and how do they relate to vertical angles?
5. How could you confirm this solution using a geometric proof?

**Tip:** Always remember that vertical angles, formed by the intersection of two straight lines, are congruent—this is a fundamental geometric property.

Find the value of the variable and the measure of each labeled angle in the system of vertical angles: (x + 17)° and (4x - 34)°.

This page details how to solve for the variable x in a system of vertical angles using algebra. The problem involves setting the angles equal and solving for x to find that both angles measure 34°. Perfect for students in grades 7-9.

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