https://file.aiquickdraw.com/mathbot/file/17264277953678kcqeg5l.jpg

The image shows a diagram with three labeled angles: \( \angle 1 \), \( \angle 2 \), and \( \angle 3 \). The given information is that one of the angles, adjacent to \( \angle 1 \), measures \( 22^\circ \).

Here are the steps to solve for the measures of \( \angle 1 \), \( \angle 2 \), and \( \angle 3 \):

1. **Vertical Angles**: Vertical angles are always congruent (equal in measure). In the diagram, \( \angle 1 \) is vertical to the \( 22^\circ \) angle, so:
   \[
   m\angle 1 = 22^\circ
   \]

2. **Linear Pair**: Adjacent angles on a straight line add up to \( 180^\circ \) (this is called a linear pair). \( \angle 1 \) and \( \angle 2 \) form a linear pair, so:
   \[
   m\angle 1 + m\angle 2 = 180^\circ
   \]
   Substituting \( m\angle 1 = 22^\circ \):
   \[
   22^\circ + m\angle 2 = 180^\circ
   \]
   Solving for \( m\angle 2 \):
   \[
   m\angle 2 = 180^\circ - 22^\circ = 158^\circ
   \]

3. **Vertical Angle for \( \angle 3 \)**: Similarly, \( \angle 2 \) and \( \angle 3 \) are vertical angles, so they must be equal. Therefore:
   \[
   m\angle 3 = m\angle 2 = 158^\circ
   \]

### Final Answers:
- \( m\angle 1 = 22^\circ \)
- \( m\angle 2 = 158^\circ \)
- \( m\angle 3 = 158^\circ \)

Let me know if you'd like further details or have any questions!

Here are five related questions to explore:
1. What are the properties of vertical angles, and why are they always congruent?
2. How do adjacent angles form a linear pair, and how does this help in solving for unknown angles?
3. What happens if the sum of angles on a straight line is more than or less than \( 180^\circ \)?
4. Can you identify other types of angle pairs, such as complementary or supplementary angles, in diagrams like this?
5. How would the solution change if \( \angle 1 \) was given instead of \( 22^\circ \)?

**Tip**: Remember that vertical angles are always congruent, and adjacent angles on a straight line sum to \( 180^\circ \), which are key properties in geometry problems involving intersecting lines.

Find the measures of ∠1, ∠2, and ∠3 based on the diagram, where ∠1 is vertical to a 22° angle and ∠2 forms a linear pair with ∠1.

Learn how to find the measures of angles ∠1, ∠2, and ∠3 in a diagram where vertical angles and linear pairs are involved. Given that one of the angles is 22°, apply geometric theorems to solve for the other angles. Suitable for middle school students in Grades 6-8.

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