Question

which of the following statements and justifications would prove that r // s? answer attempt 1 out of 2 ∠1≅∠9, by converse of alternate exterior ∠1≅∠4, by converse of same - side interior angles ∠1≅∠5, by converse of corresponding angles ∠1≅∠15, by converse of alternate exterior angles

Explanation:

Step1: Recall parallel - line theorems

The converse of corresponding angles postulate states that if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. In the given figure, when line \(t\) is the transversal for lines \(r\) and \(s\), \(\angle1\) and \(\angle5\) are corresponding angles.

Step2: Analyze each option

• For \(\angle1\cong\angle9\), \(\angle1\) and \(\angle9\) are not related in a way (alternate - exterior, corresponding, etc.) that can prove \(r\parallel s\) directly.
• \(\angle1\) and \(\angle4\) are vertical angles, and the converse of same - side interior angles has nothing to do with vertical angles, so this cannot prove \(r\parallel s\).
• For \(\angle1\cong\angle5\), by the converse of corresponding angles, we can prove \(r\parallel s\).
• \(\angle1\) and \(\angle15\) are not in a relationship (alternate - exterior, etc.) that can prove \(r\parallel s\) for lines \(r\) and \(s\).

Answer:

\(\angle1\cong\angle5\), by Converse of Corresponding Angles