Question

given; \\(\overline{bc} \cong \overline{dc}\\); \\(\overline{ac} \cong \overline{ec}\\)\
prove: \\(\triangle bca \cong \triangle dce\\)\
\

statements	reasons	\
---	---	\
1. . .	1.	\
2.	2.	\
3.	3.	\
4.	4.

Explanation:

Step1: Identify Given Information

We know \(\overline{BC} \cong \overline{DC}\) and \(\overline{AC} \cong \overline{EC}\) (assuming a typo, should be \(\overline{EC}\) for \(\triangle DCE\)). Also, \(\angle BCA\) and \(\angle DCE\) are vertical angles, so they are congruent.

Step2: List Statements and Reasons

1. Statement 1: \(\overline{BC} \cong \overline{DC}\)

Reason 1: Given

1. Statement 2: \(\overline{AC} \cong \overline{EC}\)

Reason 2: Given (corrected from \(\overline{AC} \cong \overline{BC}\) to match \(\triangle DCE\))

1. Statement 3: \(\angle BCA \cong \angle DCE\)

Reason 3: Vertical Angles Theorem

1. Statement 4: \(\triangle BCA \cong \triangle DCE\)

Reason 4: SAS (Side - Angle - Side) Congruence Postulate

Answer:

The completed two - column proof is as follows:

Statements	Reasons
2. \(\overline{AC} \cong \overline{EC}\)	2. Given
3. \(\angle BCA \cong \angle DCE\)	3. Vertical Angles Theorem
4. \(\triangle BCA \cong \triangle DCE\)	4. SAS Congruence Postulate