https://file.aiquickdraw.com/mathbot/file/1727357402718xj0h8agi.jpg

Let's break down the proof provided in the image.

### Step 1: Understand the Diagram and the Given Information
- **Lines \( a \) and \( b \)** are parallel.
- **Line \( c \)** is the transversal cutting through both parallel lines \( a \) and \( b \).
- The goal is to show that **corresponding angles** formed by the transversal and the parallel lines are congruent. Specifically, you want to prove \( \angle 1 \cong \angle 5 \).

### Part A: **Missing Statement in Step 2**
- In Step 2, we are building the connection between angles \( 4 \) and \( 5 \), using vertical angles and the congruency of angles formed by the transversal.
- **Missing Statement**: \( \angle 2 \cong \angle 6 \). These are corresponding angles when line \( c \) cuts parallel lines \( a \) and \( b \). This fact is essential to move forward in the proof.

### Part B: **Missing Reason in Step 3**
- Step 3 states that \( \angle 4 \cong \angle 5 \).
- **Missing Reason**: **Alternate interior angles theorem**. According to this theorem, when a transversal cuts two parallel lines, alternate interior angles (like \( \angle 4 \) and \( \angle 5 \)) are congruent.

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### Recap:

- **Part A (Missing statement in Step 2)**: \( \angle 2 \cong \angle 6 \) (Corresponding angles theorem).
- **Part B (Missing reason in Step 3)**: Alternate interior angles theorem.

Would you like further details, or do you have any questions about the concepts?

### Relative questions:
1. What are corresponding angles, and why are they congruent when parallel lines are cut by a transversal?
2. Can you explain the alternate interior angles theorem in more depth?
3. How does the vertical angles theorem help in angle congruence proofs?
4. Why are transitive properties important in geometric proofs?
5. Can you explain why parallel lines and transversals play a crucial role in angle congruence?

### Tip:
Always remember to label and track angles correctly in geometry proofs to ensure each step logically follows the previous ones.

Which is the missing statement in step 2, and which is the missing reason in step 3 for a proof showing corresponding angles are congruent when two parallel lines are cut by a transversal?

Understand the proof showing that corresponding angles are congruent when two parallel lines are cut by a transversal. Explore theorems like the Alternate Interior Angles Theorem and Corresponding Angles Postulate, with step-by-step reasoning and transitive properties.

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