Question

1. lines v and w are parallel. which statement can you not deduce from this information? m∠3 + m∠5 = 180 ∠3 ≅ ∠8 ∠4 ≅ ∠5 m∠2 + m∠8 = 180

Explanation:

Step1: Recall parallel - line angle relationships

When two parallel lines \(v\) and \(w\) are cut by a transversal:

• Alternate - interior angles are congruent. So, \(\angle3\cong\angle5\) and \(\angle4\cong\angle6\).
• Alternate - exterior angles are congruent. So, \(\angle1\cong\angle7\) and \(\angle2\cong\angle8\).
• Same - side interior angles are supplementary. So, \(m\angle3 + m\angle6=180^{\circ}\) and \(m\angle4 + m\angle5 = 180^{\circ}\).
• Corresponding angles are congruent. So, \(\angle1\cong\angle5\), \(\angle2\cong\angle6\), \(\angle3\cong\angle7\), and \(\angle4\cong\angle8\).

Step2: Analyze each option

• Option 1: \(\angle3\) and \(\angle5\) are alternate - interior angles. When \(v\parallel w\), \(\angle3\cong\angle5\), not \(m\angle3 + m\angle5=180^{\circ}\).
• Option 2: \(\angle3\) and \(\angle7\) are corresponding angles. When \(v\parallel w\), \(\angle3\cong\angle7\).
• Option 3: \(\angle4\) and \(\angle5\) are same - side interior angles. When \(v\parallel w\), \(m\angle4 + m\angle5 = 180^{\circ}\), and also, since vertical angles are congruent and using parallel - line properties, we can show relationships. But \(\angle4\) and \(\angle5\) are not congruent in general for parallel lines cut by a transversal.
• Option 4: \(\angle2\) and \(\angle8\) are alternate - exterior angles. When \(v\parallel w\), \(\angle2\cong\angle8\), and \(m\angle2 + m\angle8

eq180^{\circ}\) (they are congruent, not supplementary).

Answer:

\(m\angle3 + m\angle5 = 180^{\circ}\)