Question

choose all of the following statements that prove that e and f are parallel.
draw
m∠4 + m∠6 = 180°
∠1 ≅ ∠4
∠3 ≅ ∠6
∠7 ≅ ∠8
m∠3 + m∠5 = 90°

Explanation:

Step1: Recall parallel - line postulates

If two lines are cut by a transversal, then:

1. Same - side interior angles: If the sum of same - side interior angles is 180°, the lines are parallel. \(\angle4\) and \(\angle6\) are same - side interior angles. Since \(m\angle4 + m\angle6=180^{\circ}\), lines \(E\) and \(F\) are parallel.
2. Alternate interior angles: If alternate interior angles are congruent, the lines are parallel. \(\angle3\) and \(\angle6\) are alternate interior angles. Since \(\angle3\cong\angle6\), lines \(E\) and \(F\) are parallel.
3. Vertical angles: \(\angle1\cong\angle4\) and \(\angle7\cong\angle8\) are just vertical - angle relationships and do not prove \(E\) and \(F\) are parallel. \(\angle1\) and \(\angle4\) are vertical angles, and \(\angle7\) and \(\angle8\) are vertical angles. They are always congruent regardless of whether \(E\) and \(F\) are parallel or not.
4. \(m\angle3 + m\angle5 = 90^{\circ}\) does not relate to any parallel - line postulates.

Answer:

\(m\angle4 + m\angle6 = 180^{\circ}\), \(\angle3\cong\angle6\)