Question

given: abcd is a rhombus and \\(\overline{fc} \cong \overline{ec}\\).
prove: \\(\overline{dg} \cong \overline{bg}\\).

step | statement | reason
1 | abcd is a rhombus; \\(\overline{fc} \cong \overline{ec}\\) | given
2 | \\(\angle c \cong \angle c\\) | reflexive property
3 | \\(\overline{dc} \cong \overline{bc}\\) | all sides of a rhombus / square are congruent
4 | \\(\triangle dec \cong \triangle bfc\\) | sas
5 | \\(\angle cde \cong \angle cbf\\) | select a reason...

(note: de, fb, bc and dc are segments. there is a diagram of rhombus abcd with points e, f, g as shown.)

Explanation:

Step 1: Recall Congruent Triangle Properties

When two triangles are congruent (like \( \triangle DEC \cong \triangle BFC \) from Step 4), their corresponding parts (angles and sides) are congruent. This is the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem.

Step 2: Apply CPCTC to \( \triangle DEC \) and \( \triangle BFC \)

Since \( \triangle DEC \cong \triangle BFC \), the corresponding angles \( \angle CDE \) and \( \angle CBF \) must be congruent. So the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)