Here's how to find the values of $x$ and $y$: Step 1: Find the value of $y$. The angle between a tangent ($\overline{BE}$) and a chord ($\overline{AB}$) through the point of contact ($B$) is equal to the angle in the alternate segment ($\angle ACB$). This is the Alternate Segment Theorem. Given $\angle ABF = 50^\circ$. Therefore, $$ \angle ACB = 50^\circ $$ We are given that $\overline{AC} \parallel \overline{BE}$. Since $\overline{BC}$ is a transversal intersecting these parallel lines, the alternate interior angles are equal. So, $\angle CBF = \angle ACB$. We are given $\angle CBF = y^\circ$. Therefore, $$ y = \angle ACB = 50 $$ $$ \boxed{y = 50} $$ Step 2: Find the value of $x$. Since $\overline{AC} \parallel \overline{BE}$ and $\overline{AB}$ is a transversal, the alternate interior angles are equal. So, $\angle CAB = \angle ABE$. The angle $\angle ABE$ is the angle between the tangent $\overline{BE}$ and the chord $\overline{AB}$, which is $\angle ABF = 50^\circ$. Therefore, $$ \angle CAB = 50^\circ $$ Angles subtended by the same arc ($\overline{BC}$) at the circumference are equal. Both $\angle BFC$ and $\angle BAC$ are subtended by arc $\overline{BC}$. Therefore, $$ \angle BFC = \angle BAC = 50^\circ $$ From the diagram, $D$ is a point on the line $BC$ extended, meaning $B$, $C$, and $D$ are collinear. Thus, $\angle DFC$ is the same angle as $\angle BFC$. We are given $\angle DFC = x^\circ$. Therefore, $$ x = \angle BFC = 50 $$ $$ \boxed{x = 50} $$