Question

e is the midpoint of ac
given
ae ≅ ec
reason:
select reason
be ≅ ed
reason:
select reason
∠bea ≅ ∠ced
reason:
select reason
△abe ≅ △cde
reason:
select reason

Explanation:

Step1: Reason for \( \overline{AE}\cong\overline{EC} \)

Since \( E \) is the mid - point of \( \overline{AC} \), by the definition of a mid - point, a mid - point divides a line segment into two congruent segments. So the reason is "Definition of mid - point".

Step2: Reason for \( \overline{BE}\cong\overline{ED} \)

Assume that there is some given information (not shown in the problem statement but in a typical geometric proof setup where if we have a parallelogram or some other construction where these segments are made congruent by given conditions. If we assume it's given in the problem's context (since it's a proof - building exercise), the reason is "Given".

Step3: Reason for \( \angle BEA\cong\angle CED \)

These are vertical angles. Vertical angles are formed when two lines intersect. The measure of vertical angles is equal. So the reason is "Vertical angles theorem".

Step4: Reason for \( \triangle ABE\cong\triangle CDE \)

We have \( \overline{AE}\cong\overline{EC} \), \( \overline{BE}\cong\overline{ED} \), and \( \angle BEA\cong\angle CED \). By the Side - Angle - Side (SAS) congruence criterion (if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), the reason is "SAS".

Answer:

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