Question

1. what additional information would you need to conclude that \overleftrightarrow{de} is the perpendicular bisector of \overline{ac}?

\overline{ab}\cong\overline{ad}
\overline{ab}\cong\overline{bc}
\overline{db}\cong\overline{be}
\overline{ac}\cong\overline{de}

Explanation:

Step1: Recall the definition of a perpendicular bisector

A perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two equal parts.

Step2: Analyze each option

• For $\overline{AB}\cong\overline{AD}$: This only relates two segments ($AB$ and $AD$) and has no bearing on whether $DE$ bisects $AC$.
• For $\overline{AB}\cong\overline{BC}$: If $\overline{AB}\cong\overline{BC}$, then $B$ is the mid - point of $\overline{AC}$. Since we already know that $DE\perp AC$ (from the right - angle symbol at $B$), if $B$ is the mid - point of $AC$ (because $\overline{AB}\cong\overline{BC}$), then by the definition of a perpendicular bisector, $DE$ is the perpendicular bisector of $\overline{AC}$.
• For $\overline{DB}\cong\overline{BE}$: This only relates two segments ($DB$ and $BE$) and has no bearing on whether $DE$ bisects $AC$.
• For $\overline{AC}\cong\overline{DE}$: This only relates two segments ($AC$ and $DE$) and has no bearing on whether $DE$ bisects $AC$.

Answer:

$\overline{AB}\cong\overline{BC}$