Question

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$\overline{hi}\perp\overline{gj}$ and $\angle igj\cong\angle hgj$. complete the proof that $\overline{gi}\cong\overline{gh}$.
1 $\overline{hi}\perp\overline{gj}$
2 $\angle igj\cong\angle hgj$
3 $\angle gjh\cong\angle gji$
4 $\overline{gj}\cong\overline{gj}$
5 $\triangle ghj\cong\triangle gij$
6 $\overline{gi}\cong\overline{gh}$

Explanation:

Step1: Definition of perpendicular lines

If \(\overline{HI}\perp\overline{GJ}\), then \(\angle GJH = 90^{\circ}\) and \(\angle GJI=90^{\circ}\), so \(\angle GJH\cong\angle GJI\) (right - angle congruence theorem).

Step2: Given information

\(\angle IGJ\cong\angle HGJ\) (given).

Step3: Reflexive property of congruence

For any segment \(\overline{AB}\), \(\overline{AB}\cong\overline{AB}\). So, \(\overline{GJ}\cong\overline{GJ}\) (reflexive property).

Step4: ASA (Angle - Side - Angle) congruence criterion

In \(\triangle GHJ\) and \(\triangle GIJ\), we have \(\angle HGJ\cong\angle IGJ\) (given), \(\overline{GJ}\cong\overline{GJ}\) (reflexive property), \(\angle GJH\cong\angle GJI\) (right - angle congruence). By ASA, \(\triangle GHJ\cong\triangle GIJ\).

Step5: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle GHJ\cong\triangle GIJ\), then \(\overline{GI}\cong\overline{GH}\) (CPCTC).

Answer:

1. Given (for \(\overline{HI}\perp\overline{GJ}\))
2. Given (for \(\angle IGJ\cong\angle HGJ\))
3. Definition of perpendicular (for \(\angle GJH\cong\angle GJI\))
4. Reflexive property of congruence (for \(\overline{GJ}\cong\overline{GJ}\))
5. ASA (Angle - Side - Angle) (for \(\triangle GHJ\cong\triangle GIJ\))
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent) (for \(\overline{GI}\cong\overline{GH}\))