Question

r is a transversal.

Prove: ∠2 ≅ ∠7

Parallel lines p and q are cut by transversal r. On line p where it intersects with line r, 4 angles are created. Labeled clockwise, from the uppercase left, the angles are: 1, 2, 4, 3. On line q where it intersects with line r, 4 angles are created. Labeled clockwise, from the uppercase left, the angles are: 5, 6, 8, 7.
A 2-column table with 4 rows. Column 1 is labeled statements with the entries p is parallel to q and r is a transversal, A, B, angle 2 is congruent to angle 7. Column 2 is labeled reasons with the entries given, vertical angles are congruent, correlated angle theorem, transitive property.

Which statements could complete the proof?

A:


B:

Answer 1

To complete the proof that ∠2 ≅ ∠7 with the given information, we can fill in the blanks as follows:

### Statements
A: ∠2 ≅ ∠6 (alternate interior angles)
B: ∠6 ≅ ∠7 (vertical angles are congruent)

### Reasons
A: Alternate Interior Angles Theorem
B: Vertical Angles Theorem

### Completed Proof Steps:

| Statements | Reasons |
|-------------------------|-------------------------------------------|
| 1. p is parallel to q | Given |
| 2. r is a transversal | Given |
| 3. ∠2 ≅ ∠6 | Alternate Interior Angles Theorem |
| 4. ∠6 ≅ ∠7 | Vertical Angles Theorem |
| 5. ∠2 ≅ ∠7 | Transitive Property |

In this proof, step 3 shows that ∠2 and ∠6 are congruent because they are alternate interior angles. In step 4, we state that ∠6 and ∠7 are congruent because they are vertical angles. Finally, in step 5, we conclude that ∠2 is congruent to ∠7 by using the transitive property, as both angles are congruent to ∠6.