Question

are $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ parallel? explain your reasoning.
yes; by the linear pair postulate, $mangle ebc = 65^{circ}$. so, $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ are parallel because corresponding angles are congruent.
yes; by the vertical angles congruence theorem, $mangle ebc = 115^{circ}$. so, consecutive interior angles are supplementary, and $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ are parallel by the consecutive interior angles converse.
no; because the alternate exterior angles are not marked congruent, you cannot conclude that $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ are parallel.
no; because the alternate interior angles are not marked supplementary, you cannot conclude that $overleftrightarrow{ac}$ and $overleftrightarrow{df}$ are parallel.

Explanation:

Step1: Identify angle - related theorem

We know that vertical angles are congruent. The angle opposite the $115^{\circ}$ angle is $\angle EBC$. By the Vertical Angles Congruence Theorem, $m\angle EBC = 115^{\circ}$.

Step2: Analyze consecutive - interior angles

The $115^{\circ}$ angle and the $65^{\circ}$ angle on the same - side of the transversal are consecutive interior angles. Since $115^{\circ}+65^{\circ}=180^{\circ}$, consecutive interior angles are supplementary.

Step3: Apply the converse theorem

According to the Consecutive Interior Angles Converse, if consecutive interior angles are supplementary, then the two lines are parallel. So, $\overleftrightarrow{AC}$ and $\overleftrightarrow{DF}$ are parallel.

Answer:

yes; By the Vertical Angles Congruence Theorem, $m\angle EBC = 115^{\circ}$. So, consecutive interior angles are supplementary, and $\overleftrightarrow{AC}$ and $\overleftrightarrow{DF}$ are parallel by the Consecutive Interior Angles Converse.