Question

e is the mid - point of $overline{ac}$
reason:
given
$overline{ae}congoverline{ec}$
reason:
select reason
$overline{be}congoverline{ed}$
reason:
select reason
$angle beacongangle ced$
reason:
select reason
$\triangle abecong\triangle cde$
reason:
select reason

Explanation:

Step1: Define mid - point property

If \(E\) is the mid - point of \(\overline{AC}\), then by the definition of a mid - point, the two segments formed are congruent. So, \(\overline{AE}\cong\overline{EC}\).

Step2: Assume given congruence

Let's assume \(\overline{BE}\cong\overline{ED}\) is given (as no other information about how this is derived is provided).

Step3: Vertical angles are congruent

\(\angle BEA\) and \(\angle CED\) are vertical angles. By the vertical - angles theorem, vertical angles are congruent, so \(\angle BEA\cong\angle CED\).

Step4: Prove triangle congruence

We have two pairs of congruent sides (\(\overline{AE}\cong\overline{EC}\) and \(\overline{BE}\cong\overline{ED}\)) and the included angles between them are congruent (\(\angle BEA\cong\angle CED\)). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle ABE\cong\triangle CDE\).

Answer:

• For \(\overline{AE}\cong\overline{EC}\), Reason: Definition of mid - point.
• For \(\overline{BE}\cong\overline{ED}\), Reason: Given.
• For \(\angle BEA\cong\angle CED\), Reason: Vertical angles theorem.
• For \(\triangle ABE\cong\triangle CDE\), Reason: SAS (Side - Angle - Side) congruence criterion.