Question

given e is the midpoint of \\(\overline{bd}\\), complete the flowchart proof below.

image of a geometric figure with triangles and segments, and a flowchart proof with boxes: \e is the midpoint of \\(\overline{bd}\\) (reason: given)\, then three boxes: \\\(\overline{be} \cong \overline{ed}\\) (reason: select reason)\, \\\(\overline{ae} \cong \overline{ec}\\) (reason: select reason)\, \\\(\angle bea \cong \angle ced\\) (reason: select reason)\, leading to a box: \\\(\triangle abe \cong \triangle cde\\) (reason: select reason)\

Explanation:

Step1: Define midpoint property

A midpoint divides a segment into two congruent parts. So from $E$ being the midpoint of $\overline{BD}$, we get $\overline{BE} \cong \overline{ED}$.

Step2: Identify marked congruent segments

From the diagram, $\overline{AE}$ and $\overline{EC}$ are marked as congruent, so $\overline{AE} \cong \overline{EC}$ (Reason: Given marked congruence).

Step3: Identify vertical angles property

Vertical angles formed by intersecting lines are congruent. $\angle BEA$ and $\angle CED$ are vertical angles, so $\angle BEA \cong \angle CED$.

Step4: Apply triangle congruence rule

We have two pairs of congruent sides and the included congruent angle (SAS: Side-Angle-Side), so $\triangle ABE \cong \triangle CDE$.

Answer:

1. For $\boldsymbol{\overline{BE} \cong \overline{ED}}$: Reason is Definition of a Midpoint
2. For $\boldsymbol{\overline{AE} \cong \overline{EC}}$: Reason is Given (marked congruent segments)
3. For $\boldsymbol{\angle BEA \cong \angle CED}$: Reason is Vertical Angles Theorem
4. For $\boldsymbol{\triangle ABE \cong \triangle CDE}$: Reason is SAS (Side-Angle-Side) Congruence Postulate