Question
Given: m || n and p is a transversal
Prove: mAngle2 = mAngle7
Horizontal and parallel lines m and n are cut by transversal p. On line m where it intersects line p, 4 angles are created. Labeled clockwise, from uppercase left, the angles are: 1, 2, 4, 3. On line n where it intersects line p, 4 angles are created. Labeled clockwise, from uppercase left, the angles are 5, 6, 8, 7.
What is the missing reason in the proof?
Statement Reason
1. m || n and p is a transversal 1. given
2. ∠2 ≅ ∠3 2. ver. ∠s theorem
3. m∠2 = m∠3 3. def. of congruent
4. ∠3 ≅ ∠7 4. corr. ∠s theorem
5. m∠3 = m∠7 5. def. of congruent
6. m∠2 = m∠7 6. ?
corresponding angles theorem
alternate interior angles theorem
transitive property
subtraction property
Prove: mAngle2 = mAngle7
Horizontal and parallel lines m and n are cut by transversal p. On line m where it intersects line p, 4 angles are created. Labeled clockwise, from uppercase left, the angles are: 1, 2, 4, 3. On line n where it intersects line p, 4 angles are created. Labeled clockwise, from uppercase left, the angles are 5, 6, 8, 7.
What is the missing reason in the proof?
Statement Reason
1. m || n and p is a transversal 1. given
2. ∠2 ≅ ∠3 2. ver. ∠s theorem
3. m∠2 = m∠3 3. def. of congruent
4. ∠3 ≅ ∠7 4. corr. ∠s theorem
5. m∠3 = m∠7 5. def. of congruent
6. m∠2 = m∠7 6. ?
corresponding angles theorem
alternate interior angles theorem
transitive property
subtraction property
Answers
Answered by
GPT-4o mini
To prove that \( m\angle 2 = m\angle 7 \) given the angles and their relationships through parallel lines cut by a transversal, we can use the transitive property of equality. Here's how the proof would look with the missing reason filled in:
| Statement | Reason |
|-----------|--------|
| 1. \( m \parallel n \) and \( p \) is a transversal | 1. Given |
| 2. \( \angle 2 \cong \angle 3 \) | 2. Vertical angles theorem |
| 3. \( m\angle 2 = m\angle 3 \) | 3. Definition of congruence |
| 4. \( \angle 3 \cong \angle 7 \) | 4. Corresponding angles theorem |
| 5. \( m\angle 3 = m\angle 7 \) | 5. Definition of congruence |
| 6. \( m\angle 2 = m\angle 7 \) | 6. Transitive property |
Thus, the missing reason in the proof is **transitive property**. This property allows us to conclude that if \( m\angle 2 = m\angle 3 \) and \( m\angle 3 = m\angle 7 \), then it follows that \( m\angle 2 = m\angle 7 \).
| Statement | Reason |
|-----------|--------|
| 1. \( m \parallel n \) and \( p \) is a transversal | 1. Given |
| 2. \( \angle 2 \cong \angle 3 \) | 2. Vertical angles theorem |
| 3. \( m\angle 2 = m\angle 3 \) | 3. Definition of congruence |
| 4. \( \angle 3 \cong \angle 7 \) | 4. Corresponding angles theorem |
| 5. \( m\angle 3 = m\angle 7 \) | 5. Definition of congruence |
| 6. \( m\angle 2 = m\angle 7 \) | 6. Transitive property |
Thus, the missing reason in the proof is **transitive property**. This property allows us to conclude that if \( m\angle 2 = m\angle 3 \) and \( m\angle 3 = m\angle 7 \), then it follows that \( m\angle 2 = m\angle 7 \).