Question

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

step	statement	reason
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\\)	select a reason...

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

Explanation:

Step1: Identify congruent parts

We have \( \overline{FC} \cong \overline{EC} \) (given), \( \angle C \cong \angle C \) (reflexive), and \( \overline{DC} \cong \overline{BC} \) (sides of rhombus).

Step2: Apply SAS congruence

In \( \triangle DEC \) and \( \triangle BFC \), two sides and the included angle are congruent: \( \overline{DC} \cong \overline{BC} \), \( \angle C \cong \angle C \), \( \overline{EC} \cong \overline{FC} \). By the Side - Angle - Side (SAS) Congruence Postulate, \( \triangle DEC \cong \triangle BFC \).

Answer:

The reason for \( \triangle DEC \cong \triangle BFC \) is the Side - Angle - Side (SAS) Congruence Postulate.