Question

select the correct answer from each drop - down menu. in the figure below, u || v. prove: ∠3 and ∠5 are congruent. statements reasons 1. u || v given 2. ∠1 ≅ ∠7 3. ∠5 ≅ ∠7 vertical angles theorem 4. ∠1 ≅ ∠3 vertical angles theorem 5. ∠3 ≅ ∠5

Explanation:

Step1: Identify angle - pair relationship

Since \(u\parallel v\), \(\angle1\) and \(\angle7\) are alternate - exterior angles.

Step2: State the reason for congruence

The reason for \(\angle1\cong\angle7\) is the alternate - exterior angles theorem which states that when two parallel lines are cut by a transversal, alternate - exterior angles are congruent.

Step3: Use transitive property

We know \(\angle1\cong\angle3\) (vertical angles theorem) and \(\angle5\cong\angle7\) (vertical angles theorem) and \(\angle1\cong\angle7\). By the transitive property of congruence (if \(a = b\) and \(b = c\), then \(a = c\)), \(\angle3\cong\angle5\). The reason for \(\angle3\cong\angle5\) is the transitive property of congruence.

Answer:

1. Reason: alternate - exterior angles theorem
2. Reason: transitive property of congruence