Question

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$\overline{hi} \perp \overline{gj}$ and $\angle igj \cong \angle hgj$. complete the proof that $\overline{gh} \cong \overline{gi}$.
1 $\overline{hi} \perp \overline{gj}$ given
2 $\angle igj \cong \angle hgj$ given
3 $\angle gjh \cong \angle gji$ all right angles are congruent
4 $\overline{gj} \cong \overline{gj}$ reflexive property of congruence
5 $\triangle ghj \cong \triangle gij$
6 $\overline{gh} \cong \overline{gi}$

Explanation:

Step1: Determine the congruence criterion

We have two angles ($\angle IGJ\cong\angle HGJ$ and $\angle GJH\cong\angle GJI$) and the included side ($\overline{GJ}\cong\overline{GJ}$). So, by the Angle - Side - Angle (ASA) congruence criterion, $\triangle GHJ\cong\triangle GIJ$.

Step2: Use the property of congruent triangles

If two triangles are congruent, then their corresponding parts are congruent. Since $\triangle GHJ\cong\triangle GIJ$, the corresponding sides $\overline{GH}$ and $\overline{GI}$ are congruent.

Answer:

For statement 5, the reason is "Angle - Side - Angle (ASA) Congruence Criterion". For statement 6, the reason is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".