Select the correct answer from each drop-down menu.

**Given:** a∥b

**Prove:** ∠1≅ ∠7

Complete the proof.

StatementsReasons

1. a∥b

given

2. ∠1≅ ∠5

corresponding angles theorem

3. ∠512 ≅ ∠7

vertical angles theorem

4. ∠1≅ ∠487

transitive property of congruence


To complete the proof that \( \angle 1 \cong \angle 7 \) given that lines \( a \parallel b \), let's analyze the statements and reasons provided and fill in the missing information.

### Proof:

Given: \( a \parallel b \)  
To Prove: \( \angle 1 \cong \angle 7 \)

**Statements and Reasons:**

1. **Statement:** \( a \parallel b \)  
   **Reason:** Given

2. **Statement:** \( \angle 1 \cong \angle 5 \)  
   **Reason:** Corresponding Angles Postulate (angles that are in the same relative position when a transversal crosses two parallel lines are congruent).

3. **Statement:** \( \angle 5 \cong \angle 7 \)  
   **Reason:** Vertical Angles Theorem (vertical angles are congruent).

4. **Statement:** \( \angle 1 \cong \angle 7 \)  
   **Reason:** Transitive Property of Congruence (if \( \angle 1 \cong \angle 5 \) and \( \angle 5 \cong \angle 7 \), then \( \angle 1 \cong \angle 7 \)).

### Correct Choices for the Drop-Down Menu:

- Corresponding angles **theorem** (Step 2 Reason)
- Vertical angles **theorem** (Step 3 Reason)
- Transitive property of **congruence** (Step 4 Reason)

### Completed Proof:

1. **a ∥ b** — given  
2. **∠1 ≅ ∠5** — corresponding angles theorem  
3. **∠5 ≅ ∠7** — vertical angles theorem  
4. **∠1 ≅ ∠7** — transitive property of congruence  

Would you like more details or have any questions? Here are some related questions that might help further your understanding:

1. What is the Corresponding Angles Postulate in geometry?
2. How do vertical angles form and why are they congruent?
3. What is the Transitive Property of Congruence in geometry?
4. How can you prove that two angles are congruent using alternate interior angles?
5. What are some other angle relationships formed by parallel lines and a transversal?

**Tip:** When working with proofs involving parallel lines and a transversal, always consider angle pairs such as corresponding angles, alternate interior angles, and vertical angles for establishing congruence.

Learn how to prove angle congruence using parallel lines and a transversal in geometry. This detailed proof covers the corresponding angles postulate, vertical angles theorem, and the transitive property of congruence. Suitable for high school students studying geometry.

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