Question

fill in the missing statement and reason
the proof below.
given: \\( \overline { b d } \\) and \\( \overline { a c } \\) bisect each other.
prove: \\( \triangle a b e \cong \triangle c d e \\).

Explanation:

Step1: Recall the property of bisecting segments

Since \(\overline{BD}\) and \(\overline{AC}\) bisect each other, by the definition of bisecting segments, if two segments bisect each other, the segments are divided into two equal - length sub - segments. So, if \(\overline{BD}\) and \(\overline{AC}\) bisect each other at point \(E\), then \(\overline{BE}\cong\overline{ED}\) (because when two segments bisect each other, the parts of one segment are congruent to the corresponding parts of the other segment).

Step2: Check the proof structure

We are using the Side - Angle - Side (SAS) congruence criterion to prove \(\triangle ABE\cong\triangle CDE\). In step 2, we have \(\overline{AE}\cong\overline{EC}\) (from the bisecting property), in step 4, we have \(\angle AEB\cong\angle CED\) (vertical angles are congruent). For the SAS criterion, we need two sides and the included angle. We already have one pair of sides (\(\overline{AE}\cong\overline{EC}\)) and one pair of angles (\(\angle AEB\cong\angle CED\)). The missing pair of sides for the SAS criterion is \(\overline{BE}\cong\overline{ED}\)

Answer:

The missing statement in step 3 is \(\overline{BE}\cong\overline{ED}\) and the reason is "Definition of bisecting segments"