Question

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$\angle h\cong\angle i$ and $\overline{hi}\perp\overline{gj}$. complete the proof that $\overline{gi}\cong\overline{gh}$.
1 $\angle h\cong\angle i$ given
2 $\overline{hi}\perp\overline{gj}$ 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 $\overline{gi}\cong\overline{gh}$

Explanation:

Step1: Determine the congruence criterion for triangles

We have \(\angle H\cong\angle I\) (given), \(\angle GJH\cong\angle GJI\) (all right angles are congruent), and \(\overline{GJ}\cong\overline{GJ}\) (reflexive property). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle GHJ\cong\triangle GIJ\).

Step2: Use the property of congruent triangles

If two triangles \(\triangle GHJ\) and \(\triangle GIJ\) are congruent, then their corresponding sides are congruent. Since \(\overline{GI}\) and \(\overline{GH}\) are corresponding sides of congruent triangles \(\triangle GIJ\) and \(\triangle GHJ\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\overline{GI}\cong\overline{GH}\).

Answer:

1. AAS (Angle - Angle - Side) Congruence Criterion; 6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)