Question

complete the proof that $\triangle ghi \cong \triangle gih$.

statement reason
1 $\angle h \cong \angle i$ given
2 $\overline{hi} \cong \overline{hi}$ reflexive property of congruence
3 $\triangle ghi \cong \triangle gih$

Explanation:

Step1: Identify given congruent parts

$\angle H \cong \angle I$, $\overline{HI} \cong \overline{HI}$

Step2: Match to congruence criterion

We have one pair of congruent angles, one shared congruent side, and we note that for $\triangle GHI$ and $\triangle GIH$, the side $\overline{HI}$ is between the angles: in $\triangle GHI$, $\angle H$ is adjacent to $\overline{HI}$ and $\overline{GH}$; in $\triangle GIH$, $\angle I$ is adjacent to $\overline{HI}$ and $\overline{GI}$. Wait, correction: actually, the given is $\angle H \cong \angle I$, shared side $\overline{HI}$, and we can confirm the ASA (Angle-Side-Angle) criterion: $\angle H$ (of $\triangle GHI$) $\cong \angle I$ (of $\triangle GIH$), $\overline{HI} \cong \overline{HI}$, and the included side is the shared side, so the remaining angle correspondence is implicit, but the valid congruence rule here is ASA.

Step3: State final congruence reason

$\triangle GHI \cong \triangle GIH$ by ASA Congruence Criterion

Answer:

The missing reason for statement 3 is ASA (Angle-Side-Angle) Congruence Criterion

Full completed proof:

Statement	Reason
2. $\overline{HI} \cong \overline{HI}$	Reflexive Property of Congruence
3. $\triangle GHI \cong \triangle GIH$	ASA Congruence Criterion