Question

lines a, b, c, and d intersect as shown.

which pair of lines are parallel?
a. a and b
b. a and c
c. c and d
d. b and d

Explanation:

Step1: Check Line b and d

For lines \( b \) and \( d \), the corresponding angles with transversal \( c \) should be equal if parallel. The angle at \( c \) for \( b \) is \( 95^\circ \), and for \( d \) it's also \( 95^\circ \)? Wait, no, let's recheck. Wait, the transversal for \( b \) and \( d \) is \( c \). The angle between \( b \) and \( c \) is \( 95^\circ \), and between \( d \) and \( c \) is \( 95^\circ \)? Wait, no, actually, for \( b \) and \( d \), the corresponding angles: when a transversal (here \( c \)) cuts two lines, corresponding angles should be equal. The angle above \( c \) for \( b \) is \( 95^\circ \), and below \( c \) for \( d \) is \( 95^\circ \), but they are vertical? Wait, no, maybe I made a mistake. Let's check \( b \) and \( d \): the angle between \( b \) and \( c \) is \( 95^\circ \), and between \( d \) and \( c \) is \( 95^\circ \), so they are corresponding angles? Wait, no, \( c \) is a transversal, so if \( b \parallel d \), then corresponding angles should be equal. The angle at the intersection of \( b \) and \( c \) (above \( c \)) is \( 95^\circ \), and at \( d \) and \( c \) (below \( c \)) is \( 95^\circ \), but they are vertical angles? Wait, no, maybe \( b \) and \( d \): let's check the other angles. Wait, the angle between \( b \) and the line (let's see, the triangle: \( 42 + 53 + 39 = 134 \), no, \( 42 + 53 = 95 \), \( 95 + 39 = 134 \), no. Wait, maybe \( b \) and \( d \): the angle adjacent to \( 95^\circ \) on line \( b \) is \( 180 - 95 = 85^\circ \), and on line \( d \), adjacent to \( 95^\circ \) is \( 180 - 95 = 85^\circ \)? No, that's not. Wait, maybe I'm wrong. Wait, let's check \( c \) and \( d \): are they parallel? No, because \( c \) and \( d \) are both horizontal? Wait, no, \( c \) and \( d \) are both horizontal? Wait, the diagram shows \( c \) and \( d \) as horizontal lines? Wait, no, the arrows for \( c \) and \( d \) are both horizontal (left and right), so they are parallel? Wait, no, the options include \( c \) and \( d \) as option C, but the correct answer is D? Wait, no, the user marked B, but let's re-express. Wait, the correct way: for two lines to be parallel, corresponding angles, alternate interior angles, or consecutive interior angles should satisfy the conditions. Let's check \( b \) and \( d \): the transversal is \( c \), and the corresponding angles (the \( 95^\circ \) angles) are equal (since both are \( 95^\circ \)), so \( b \parallel d \)? Wait, but the user's selected option is B, but maybe I made a mistake. Wait, no, let's check \( a \) and \( c \): the angle adjacent to \( 124^\circ \) is \( 180 - 124 = 56^\circ \). Is there a corresponding angle for \( a \) and \( c \)? Wait, maybe not. Wait, the correct answer is D: \( b \) and \( d \), because the corresponding angles (the \( 95^\circ \) angles) are equal when cut by transversal \( c \). So \( b \parallel d \), so option D.

Step2: Verify Other Options

- Option A: \( a \) and \( b \): no corresponding angles equal.
- Option B: \( a \) and \( c \): no, the angle for \( a \) is \( 124^\circ \), and for \( c \) no matching angle.
- Option C: \( c \) and \( d \): they are both horizontal? Wait, no, in the diagram, \( c \) and \( d \) are parallel? Wait, no, the angles: if \( c \parallel d \), then corresponding angles should be equal. But the angle above \( c \) for \( b \) is \( 95^\circ \), and below \( d \) is \( 95^\circ \), but \( c \) and \( d \) are horizontal, so they should be parallel? But that's not matching. Wait, I think I messed up. Let's re-express:

Wait, the key is: when a transversal cuts two lines, if corresponding angles are equal, the lines are parallel. For \( b \) and \( d \), the transversal is \( c \). The angle between \( b \) and \( c \) (above \( c \)) is \( 95^\circ \), and between \( d \) and \( c \) (below \( c \)) is \( 95^\circ \), so they are corresponding angles (since \( c \) is the transversal, same position relative to \( b \) and \( d \)), so \( b \parallel d \). So the correct answer is D.

Answer:

D. \( b \) and \( d \)