Question

yes, by the vertical angles congruence theorem, m∠feb = 123°. so, ac and df are parallel because vertical angles are congruent. yes, by the linear pair postulate, m∠deb = 57°. so, ac and df are parallel by the corresponding angles converse. no, because the consecutive exterior angles are not congruent, ac and df are not parallel. no, because the alternate exterior angles are not supplementary, ac and df are not parallel.

Explanation:

Step1: Identify angle - related postulates and theorems

We know that $\angle ABE = 57^{\circ}$ and $\angle DEF=123^{\circ}$. By the Linear - Pair Postulate, since $\angle DEB$ and $\angle DEF$ form a linear pair ($\angle DEB+\angle DEF = 180^{\circ}$), if $\angle DEF = 123^{\circ}$, then $m\angle DEB=180 - 123=57^{\circ}$.

Step2: Apply the Corresponding Angles Converse

Corresponding angles are angles in the same relative position with respect to the transversal and the two lines. $\angle ABE$ and $\angle DEB$ are corresponding angles. If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Here, $m\angle ABE = 57^{\circ}$ and $m\angle DEB = 57^{\circ}$, so $\overleftrightarrow{AC}$ and $\overleftrightarrow{DF}$ are parallel by the Corresponding Angles Converse.

Answer:

yes, By the Linear Pair Postulate, $m\angle DEB = 57^{\circ}$. So, $\overleftrightarrow{AC}$ and $\overleftrightarrow{DF}$ are parallel by the Corresponding Angles Converse.