Question

use the figures to determine whether the indicated lines are parallel. indicate why or why not. a) are m and n parallel lines? why or why not?
a. yes, because vertical angles are not congruent.
b. no, because adjacent angles are 36° each.
c. no, because opposite interior angles are supplementary.
d. yes, because alternate exterior angles are 36°.

Explanation:

To determine if lines \( m \) and \( n \) are parallel, we analyze the angles formed by the transversal. The \( 144^\circ \) angle and \( 36^\circ \) angle: first, check the relationship. The consecutive interior angles (or same - side interior angles) should be supplementary for parallel lines? Wait, no, let's look at alternate exterior angles. The \( 36^\circ \) angle and the angle adjacent to \( 144^\circ \): the angle adjacent to \( 144^\circ \) (linear pair) is \( 180 - 144=36^\circ \). So alternate exterior angles are equal (\( 36^\circ \) each). By the alternate exterior angles theorem, if alternate exterior angles are congruent, the lines are parallel. Now let's check the options:

• Option A: Vertical angles should be congruent, so this is wrong.
• Option B: The reasoning is incorrect, adjacent angles here are supplementary (\( 144 + 36 = 180\)), not \( 36^\circ \) each.
• Option C: Opposite interior angles (same - side interior) are supplementary (\( 144+36 = 180\)), which actually would imply parallel lines, but the option says "No", so it's wrong.
• Option D: Alternate exterior angles are \( 36^\circ \) (the angle adjacent to \( 144^\circ \) and the \( 36^\circ \) angle on the other line), so by alternate exterior angles theorem, lines are parallel. This is correct.

Answer:

D. Yes, because alternate exterior angles are \( 36^\circ \).