Question
Given: Lines p and q are parallel and 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:
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:
Answers
GPT-4o mini
To complete the proof that \(\angle 2 \cong \angle 7\) given the parallel lines \(p\) and \(q\) cut by transversal \(r\), you can fill in the two columns as follows:
A:
1. \(\angle 2 \cong \angle 6\) (corresponding angles)
2. \(\angle 6 \cong \angle 7\) (vertical angles are congruent)
B:
1. Corresponding Angle Postulate (if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent)
2. Vertical Angles Theorem (vertical angles are congruent)
Using these statements, we can conclude that:
\(\angle 2 \cong \angle 7\) by the Transitive Property (if \(\angle 2 \cong \angle 6\) and \(\angle 6 \cong \angle 7\), then \(\angle 2 \cong \angle 7\)).
Here’s how the final proof would look:
| Statements | Reasons |
|--------------------------------|-----------------------------------------------|
| 1. \(p \parallel q\) | Given |
| 2. \(r\) is a transversal | Given |
| 3. \(\angle 2 \cong \angle 6\) | Corresponding Angle Postulate |
| 4. \(\angle 6 \cong \angle 7\) | Vertical Angles Theorem |
| 5. \(\angle 2 \cong \angle 7\) | Transitive Property |
This proof concludes that \(\angle 2\) is congruent to \(\angle 7\) using the properties of parallel lines and the relationships between angles formed by transversals.
A:
1. \(\angle 2 \cong \angle 6\) (corresponding angles)
2. \(\angle 6 \cong \angle 7\) (vertical angles are congruent)
B:
1. Corresponding Angle Postulate (if two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent)
2. Vertical Angles Theorem (vertical angles are congruent)
Using these statements, we can conclude that:
\(\angle 2 \cong \angle 7\) by the Transitive Property (if \(\angle 2 \cong \angle 6\) and \(\angle 6 \cong \angle 7\), then \(\angle 2 \cong \angle 7\)).
Here’s how the final proof would look:
| Statements | Reasons |
|--------------------------------|-----------------------------------------------|
| 1. \(p \parallel q\) | Given |
| 2. \(r\) is a transversal | Given |
| 3. \(\angle 2 \cong \angle 6\) | Corresponding Angle Postulate |
| 4. \(\angle 6 \cong \angle 7\) | Vertical Angles Theorem |
| 5. \(\angle 2 \cong \angle 7\) | Transitive Property |
This proof concludes that \(\angle 2\) is congruent to \(\angle 7\) using the properties of parallel lines and the relationships between angles formed by transversals.