Question

statement reason
$overline{ab}paralleloverline{cd}$ given
$overline{bd}$ bisects $overline{ac}$
$\triangle abecong\triangle cde$ asa
note: $overline{db}$ and $overline{ac}$ are segments.

Explanation:

Step1: Identify vertical - angles

$\angle AEB$ and $\angle DEC$ are vertical - angles, so $\angle AEB=\angle DEC$ (Vertical angles are equal).

Step2: Use parallel - line properties

Since $\overline{AB}\parallel\overline{CD}$, then $\angle BAE=\angle DCE$ (Alternate interior angles for parallel lines $\overline{AB}$ and $\overline{CD}$ with transversal $\overline{AC}$).

Step3: Use the bisecting - segment property

$\overline{BD}$ bisects $\overline{AC}$, so $AE = CE$.

Step4: Prove triangle congruence

In $\triangle ABE$ and $\triangle CDE$, we have $\angle BAE=\angle DCE$, $AE = CE$, and $\angle AEB=\angle DEC$. By the ASA (Angle - Side - Angle) congruence criterion, $\triangle ABE\cong\triangle CDE$.

Answer:

The proof that $\triangle ABE\cong\triangle CDE$ by ASA is correct. The key steps are identifying vertical angles, using parallel - line properties for alternate interior angles, and the property of the bisecting segment.