Question

use the figures to determine whether the indicated lines are
a)

b)

d)

e)

b) are q and r parallel lines? why or why not?
a. yes, because corresponding angles are 83°
b. no, because adjacent angles are supplementary
c. yes, because a triangle fits between q and r
d. no, because opposite interior angles are congruent

Explanation:

To determine if lines \( q \) and \( r \) are parallel, we check the corresponding angles formed by the transversal \( p \). The angle on line \( q \) is \( 97^\circ \) and on line \( r \) is \( 83^\circ \). For lines to be parallel, corresponding angles should be equal (or supplementary in some cases, but here we check corresponding angles). Wait, actually, let's recast: the sum of \( 97^\circ \) and \( 83^\circ \) is \( 180^\circ \), but wait, no—wait, the angles given: the angle above line \( q \) with transversal \( p \) is \( 97^\circ \), and below line \( r \) is \( 83^\circ \). Wait, maybe I misread. Wait, the correct approach: when a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary. Let's check the sum of \( 97^\circ \) and \( 83^\circ \): \( 97 + 83 = 180 \), but those are adjacent angles? Wait, no, the angle on line \( q \) (above) is \( 97^\circ \), and the angle on line \( r \) (below) is \( 83^\circ \). Wait, actually, the angle adjacent to \( 97^\circ \) on line \( q \) would be \( 180 - 97 = 83^\circ \), which matches the angle on line \( r \). So corresponding angles (the \( 83^\circ \) angle on \( r \) and the supplementary angle of \( 97^\circ \) on \( q \)) are equal, meaning the lines are parallel? Wait, no, the option A says "Yes, because corresponding angles are \( 83^\circ \)". Let's check the options:

• Option A: Yes, because corresponding angles are \( 83^\circ \). If we consider the angle adjacent to \( 97^\circ \) on line \( q \) (which is \( 180 - 97 = 83^\circ \)) and the angle on line \( r \) is \( 83^\circ \), these are corresponding angles, so they are equal, hence lines \( q \) and \( r \) are parallel.

• Option B: No, because adjacent angles are supplementary. Adjacent angles on a straight line are supplementary, but that's always true, not a reason for non - parallelism.

• Option C: A triangle fitting between them is not a valid reason for parallelism.

• Option D: Opposite interior angles congruent is not a reason for non - parallelism (in fact, for parallel lines, alternate interior angles are congruent).

So the correct answer is A.

Answer:

A. Yes, because corresponding angles are \( 83^\circ \)