Question

i is the midpoint of \\(\overline{gh}\\) and \\(\overline{fi} \perp \overline{gh}\\). complete the proof that \\(\overline{fg} \cong \overline{fh}\\).

statement	reason
2. \\(\overline{fi} \perp \overline{gh}\\)	given
3. \\(\angle fig \cong \angle fih\\)	all right angles are congruent
4. \\(\overline{gi} \cong \overline{hi}\\)	definition of midpoint
5. \\(\overline{fi} \cong \overline{fi}\\)	reflexive property of congruence
6. \\(\triangle fgi \cong \triangle fhi\\)
7. \\(\overline{fg} \cong \overline{fh}\\)

Explanation:

Step1: Identify congruence rule

We have two pairs of congruent sides ($\overline{GI} \cong \overline{HI}$, $\overline{FI} \cong \overline{FI}$) and a pair of congruent included right angles ($\angle FIG \cong \angle FIH$). This matches the SAS (Side-Angle-Side) Congruence Postulate.

Step2: Justify corresponding parts

Once the triangles are proven congruent, their corresponding sides are congruent by the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem.

Answer:

Statement	Reason
7. $\overline{FG} \cong \overline{FH}$	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)