This image involves geometry concepts where two parallel lines $p$ and $q$ are intersected by a transversal $w$. We are given that $\angle 1 = 55^\circ$, and you are asked to complete statements regarding the relationships between the angles.

Let's break down the problem step-by-step based on transversal properties:

1. Relationship between $\angle 1$ and $\angle 2$:

   • $\angle 1$ and $\angle 2$ are alternate interior angles. Since the lines $p$ and $q$ are parallel and cut by the transversal $w$, alternate interior angles are equal. So: $\angle 1 = \angle 2 = 55^\circ$

2. Relationship between $\angle 2$ and $\angle 3$:

   • $\angle 2$ and $\angle 3$ are corresponding angles. Corresponding angles are also congruent when the transversal crosses parallel lines. Therefore: $\angle 2 = \angle 3 = 55^\circ$

3. Relationship between $\angle 1$ and $\angle 3$:

   • $\angle 1$ and $\angle 3$ are corresponding angles as well, since they occupy the same relative positions at each intersection. Thus: $\angle 1 = \angle 3 = 55^\circ$

With this information, you can fill in the blanks in the provided statements accordingly:

1. $\angle 1$ and $\angle 2$ are alternate interior angles.
2. $m \angle 2 = 55^\circ$.
3. $\angle 2$ and $\angle 3$ are corresponding angles.
4. $m \angle 3 = 55^\circ$.
5. $\angle 1$ and $\angle 3$ are corresponding angles.

Would you like further clarification, or do you want to go over any specific parts?

Here are five related questions to expand on this topic:

1. What are consecutive interior angles, and how are they related when two parallel lines are cut by a transversal?
2. If $\angle 1$ was instead $70^\circ$, how would that affect $\angle 2$ and $\angle 3$?
3. How do we prove that alternate interior angles are equal when parallel lines are cut by a transversal?
4. Can supplementary angles also occur with a transversal cutting two parallel lines? If so, which ones?
5. What would change if the lines $p$ and $q$ were not parallel?

Tip: Always remember that corresponding and alternate interior angles are congruent when a transversal cuts parallel lines.