The diagram shows two parallel lines $a$ and $b$, intersected by a transversal line $z$. We are given that $m \angle 2 = 22^\circ$, and we need to determine the measures of the other labeled angles.

Since lines $a$ and $b$ are parallel, we can use the following angle relationships:

• Corresponding Angles: Equal in measure.
• Alternate Interior Angles: Equal in measure.
• Consecutive Interior Angles: Sum to $180^\circ$.

Let's solve step-by-step:

1. $\angle 2 = 22^\circ$ (given).

2. By corresponding angles, $\angle 6$ is also $22^\circ$ because $\angle 2$ corresponds to $\angle 6$.

3. By alternate interior angles, $\angle 3 = 22^\circ$ because $\angle 2$ and $\angle 3$ are on opposite sides of the transversal.

4. By corresponding angles, $\angle 7 = 22^\circ$ because $\angle 3$ corresponds to $\angle 7$.

5. Now for $\angle 1$. It is a linear pair with $\angle 2$, so $\angle 1 = 180^\circ - 22^\circ = 158^\circ$.

6. By corresponding angles, $\angle 5 = 158^\circ$ because $\angle 1$ corresponds to $\angle 5$.

7. By alternate interior angles, $\angle 4 = 158^\circ$ because $\angle 1$ and $\angle 4$ are on opposite sides of the transversal.

8. Finally, $\angle 8 = 158^\circ$ by corresponding angles because $\angle 4$ corresponds to $\angle 8$.

Summary:

• $m\angle 1 = 158^\circ$
• $m\angle 2 = 22^\circ$ (given)
• $m\angle 3 = 22^\circ$
• $m\angle 4 = 158^\circ$
• $m\angle 5 = 158^\circ$
• $m\angle 6 = 22^\circ$
• $m\angle 7 = 22^\circ$
• $m\angle 8 = 158^\circ$

Would you like more details on any of these concepts, or have any questions?

Here are five related questions to explore further:

1. What is the difference between corresponding and alternate interior angles?
2. How would the angles change if the lines $a$ and $b$ were not parallel?
3. What other geometric properties apply when a transversal crosses parallel lines?
4. How do consecutive interior angles relate to parallel lines?
5. Can you apply these angle relationships to other polygons involving parallel lines?

Tip: Always remember that parallel lines create predictable angle patterns when intersected by a transversal, which can simplify many geometry problems.