Question

problem 3 determining whether lines are parallel
the fence gate shown below is made up of pieces of wood arranged in various directions. suppose \\( \angle 1 \cong \angle 2 \\). are lines \\( r \\) and \\( s \\) parallel? explain.
yes, \\( r \\| s \\). \\( \angle 1 \\) and \\( \angle 2 \\) are alternate exterior angles. if two lines and a transversal form congruent alternate exterior angles, then the lines are parallel (converse of the alternate exterior angles theorem).

1. as another approach which can be used with the angles 1, 2, and 3 identified to be the final reason that \\( r \\| s \\) is...

vertical angles are congruent
if corresponding angles are congruent then the lines are parallel
linear pair are supplementary
if consecutive interior angles are supplementary then the lines are parallel

Explanation:

• Vertical angles are congruent, but this property alone doesn't prove \(r\parallel s\).
• A linear pair being supplementary is about adjacent angles forming a straight - line, not relevant to proving \(r\parallel s\) here.
• Consecutive interior angles being supplementary is a different parallel - line criterion (not applicable as we are dealing with \(\angle1\) and \(\angle2\) which are not consecutive interior angles).
• If \(\angle1\cong\angle2\) and we assume \(\angle1\) and \(\angle3\) are vertical angles (so \(\angle1\cong\angle3\)), then by transitivity \(\angle2\cong\angle3\). Since \(\angle2\) and \(\angle3\) are corresponding angles, by the converse of the corresponding angles postulate (If corresponding angles are congruent then the lines are parallel), we can prove \(r\parallel s\).

Answer:

If Corresponding angles are congruent then the lines are parallel