Question

determine which lines are parallel based on the diagram below. given: ∠4≅∠5. prove: m||n statements reasons ∠4≅∠5 1. given ∠1≅∠4 2. vertical ∠s are congruent ∠1≅∠5 3. transitive property m||n 4. corresponding ∠s postulate answer choices: vertical angle theorem, corresponding angle converse, vertical angle theorem, alternate exterior angle theorem identify the three correct angles that are congruent to ∠1.

Explanation:

Step1: Recall parallel - line angle relationships

When two parallel lines are cut by a transversal, corresponding angles are congruent, alternate - interior angles are congruent, and alternate - exterior angles are congruent.

Step2: Analyze the given angles in the first part

There are no clear angle - relationships given for lines \(a\), \(b\), \(c\), \(d\) and transversal \(t\) in the first part of the problem to determine parallel lines directly from the provided angles.

Step3: Analyze the proof in the second part

1. Given \(\angle4\cong\angle5\).
2. \(\angle1\cong\angle4\) because vertical angles are congruent.
3. By the transitive property, since \(\angle1\cong\angle4\) and \(\angle4\cong\angle5\), we have \(\angle1\cong\angle5\).
4. \(\angle1\) and \(\angle5\) are corresponding angles. By the corresponding - angles postulate (if corresponding angles are congruent, then the lines are parallel), we can conclude that \(m\parallel n\).

Answer:

In the proof part, the reasons for the statements are:

1. Given
2. Vertical angles are congruent
3. Transitive property
4. Corresponding - angles postulate.

For the first part about determining parallel lines from the diagram of lines \(a\), \(b\), \(c\), \(d\) and transversal \(t\), more information about angle - relationships is needed to make a conclusion.