Question

e is the midpoint of \\( \overline { b d } \\)
reason:
given
\\( \overline { b e } \cong \overline { e d } \\)
reason:
reflexive property
\\( \angle b \cong \angle d \\)
reason:
select reason
\\( \angle a \cong \angle c \\)
reason:
select reason
\\( \triangle a b e \cong \triangle c d e \\)
reason:
select reason

Explanation:

Step1: Mid - point property

Since \(E\) is the mid - point of \(\overline{BD}\), by the definition of a mid - point, \(BE = ED\), so \(\overline{BE}\cong\overline{ED}\) (the reason for \(\overline{BE}\cong\overline{ED}\) is the definition of a mid - point, not the reflexive property. The reflexive property is for a segment being congruent to itself like \(AB\cong AB\)).

Step2: Vertical angles

If we assume that \(\angle AEB\) and \(\angle CED\) are vertical angles, then \(\angle AEB\cong\angle CED\) (vertical angles are congruent).

Step3: AAS (Angle - Angle - Side) criterion

We know that \(\angle A\cong\angle C\) (given), \(\angle B\cong\angle D\) (given), and \(\overline{BE}\cong\overline{ED}\) (from the mid - point property).
The AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Answer:

• For \(\angle B\cong\angle D\) and \(\angle A\cong\angle C\), if there is no other information given in the figure (assuming they are given as part of the problem's conditions).
• For \(\triangle ABE\cong\triangle CDE\), the reason is \(AAS\) (Angle - Angle - Side).