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.