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The problem states that \(\angle 1\) and \(\angle 2\) are vertical angles, and provides their measurements in terms of \(x\):

- \( m\angle 1 = (x + 15)^\circ \)
- \( m\angle 2 = (5x - 17)^\circ \)

Since vertical angles are congruent, we know that:
\[
m\angle 1 = m\angle 2
\]
Thus, we can set the expressions for \(m\angle 1\) and \(m\angle 2\) equal to each other:
\[
x + 15 = 5x - 17
\]

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

1. Subtract \(x\) from both sides:
\[
15 = 4x - 17
\]

2. Add 17 to both sides:
\[
32 = 4x
\]

3. Divide both sides by 4:
\[
x = 8
\]

Now that we have \(x = 8\), we can find \(m\angle 2\):

\[
m\angle 2 = 5x - 17 = 5(8) - 17 = 40 - 17 = 23^\circ
\]

Thus, the measure of \(\angle 2\) is \(23^\circ\).

Would you like more details on any part of the solution?

Here are 5 related questions to expand your understanding:

1. What are vertical angles and how are they formed?
2. How do we know that vertical angles are always congruent?
3. Can vertical angles ever have negative measurements?
4. How would the solution change if the angles were supplementary instead of vertical?
5. How can we verify that the solution for \(x\) is correct by plugging it back into both angle expressions?

**Tip:** Whenever you see vertical angles in a geometry problem, remember they are always equal, so you can set their measures equal to each other to solve for unknown variables.

∠1 and ∠2 are vertical angles. If m∠1 = (x + 15)° and m∠2 = (5x - 17)°, then find the measure of ∠2.

Learn how to solve a problem involving vertical angles where m∠1 = (x + 15)° and m∠2 = (5x - 17)°. This step-by-step solution covers algebraic manipulation and the Vertical Angles Theorem. Suitable for students in Grades 8-10.

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