Question

are $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ parallel? explain your reasoning.

143°
a b c
37°
d e f
yes; by the linear pair postulate, $mangle bef = 143^{circ}$. so, $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ are parallel by the corresponding angles converse.
yes; by the linear pair postulate, $mangle abe = 143^{circ}$. so, $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ are parallel because vertical angles are congruent.
no; because the alternate exterior angles are not congruent, $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ cannot be parallel.
no; because the alternate interior angles are not congruent, $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ cannot be parallel.

Explanation:

Step1: Recall linear - pair postulate

A linear - pair of angles is supplementary. The angle adjacent to the $37^{\circ}$ angle at point $E$ forms a linear - pair. Let the unknown angle be $\angle BEF$. Since $37^{\circ}+\angle BEF = 180^{\circ}$, then $\angle BEF=180 - 37=143^{\circ}$.

Step2: Recall corresponding - angles converse

The corresponding - angles converse states that if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. The $143^{\circ}$ angle at $B$ and $\angle BEF$ are corresponding angles. Since they are congruent ($143^{\circ}=143^{\circ}$), $\overleftrightarrow{AC}$ and $\overleftrightarrow{DF}$ are parallel.

Answer:

yes; By the Linear Pair Postulate, $m\angle BEF = 143^{\circ}$. So, $\overleftrightarrow{AC}$ and $\overleftrightarrow{DF}$ are parallel by the Corresponding Angles Converse.