Question

are ac and df parallel? explain your reasoning. 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: Use linear - pair postulate

The linear - pair postulate states that if two angles form a linear pair, they are supplementary. Given an angle of $123^{\circ}$, the angle $\angle DEB$ forms a linear pair with it. Let the given angle be $\angle DEF = 123^{\circ}$. Then $m\angle DEB=180 - 123=57^{\circ}$ since $\angle DEB+\angle DEF = 180^{\circ}$.

Step2: Apply 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. $\angle ABC = 57^{\circ}$ and $\angle DEB = 57^{\circ}$. $\angle ABC$ and $\angle DEB$ are corresponding angles. Since they are congruent, $\overleftrightarrow{AC}$ and $\overleftrightarrow{DF}$ are parallel.

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.