Question

. fill in the blanks to complete the proof below:
given: c is the midpoint of $overline{be}$ and $overline{ad}$.
prove: $\triangle abc cong \triangle dec$

statement	reason

| 1. c is midpoint of $overline{be}$ and $overline{ad}$ | 1.
| 2. $overline{bc} cong overline{ec}$ | 2. definition of a midpoint
| 3. | 3. definition of a midpoint
| 4. | 4.
| 5. $\triangle abc cong \triangle dec$ | 5.

Explanation:

Step1: Analyze Statement 3

Since \( C \) is the midpoint of \( \overline{AD} \), by the definition of a midpoint, \( \overline{AC} \cong \overline{DC} \). So the statement for step 3 is \( \overline{AC} \cong \overline{DC} \).

Step2: Analyze Statement 4

\( \angle ACB \) and \( \angle DCE \) are vertical angles. Vertical angles are congruent, so the statement is \( \angle ACB \cong \angle DCE \) and the reason is "vertical angles are congruent".

Step3: Analyze Statement 5

We have \( \overline{BC} \cong \overline{EC} \) (step 2), \( \angle ACB \cong \angle DCE \) (step 4), and \( \overline{AC} \cong \overline{DC} \) (step 3). By the Side - Angle - Side (SAS) congruence postulate, \( \triangle ABC \cong \triangle DEC \). So the reason for step 5 is "SAS congruence postulate".

Answer:

1. \( \boldsymbol{\overline{AC} \cong \overline{DC}} \)
2. \( \boldsymbol{\angle ACB \cong \angle DCE} \); Reason: vertical angles are congruent
3. Reason: SAS congruence postulate