Question

i is the midpoint of \\( \overline{g h} \\) and \\( \overline{f i} \perp \overline{g h} \\). complete the proof that \\( \triangle f g i \cong \triangle f h i \\).

statement
1 i is the midpoint of \\( \overline{g h} \\)
2 \\( \overline{f i} \perp \overline{g h} \\)
3 \\( \angle f i g \cong \angle f i h \\)
4 \\( \overline{g i} \cong \overline{h i} \\)
5 \\( \overline{f i} \cong \overline{f i} \\)
6 \\( \triangle f g i \cong \triangle f h i \\)

reason
given
given
all right angles are congruent
definition of midpoint
reflexive property of congruence
cpctc
definition of congruence

Explanation:

Step1: Use the definition of mid - point

Since \(I\) is the mid - point of \(\overline{GH}\), by the definition of mid - point, \(GI = HI\) (i.e., \(\overline{GI}\cong\overline{HI}\)).

Step2: Use the property of perpendicular lines

Given \(\overline{FI}\perp\overline{GH}\), then \(\angle FIG\) and \(\angle FIH\) are right angles. By the property that all right angles are congruent, \(\angle FIG\cong\angle FIH\).

Step3: Use the reflexive property

For the side \(\overline{FI}\), by the reflexive property of congruence, \(\overline{FI}\cong\overline{FI}\).

Step4: Apply the SAS (Side - Angle - Side) congruence criterion

In \(\triangle FGI\) and \(\triangle FHI\), we have \(\overline{GI}\cong\overline{HI}\) (from Step 1), \(\angle FIG\cong\angle FIH\) (from Step 2), and \(\overline{FI}\cong\overline{FI}\) (from Step 3). By the SAS congruence criterion (\(SAS\): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), \(\triangle FGI\cong\triangle FHI\).

Answer:

\(\triangle FGI\cong\triangle FHI\) by the \(SAS\) (Side - Angle - Side) congruence criterion.