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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.

Establishing facts about the angles created when parallel lines are cut by a transversal, given angle 1 equals 55 degrees.

This guide examines how angles are formed when parallel lines are intersected by a transversal, focusing on alternate interior and corresponding angles. The lesson is geared toward Middle School students, providing insights into foundational geometry concepts.

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