Question

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$\overline{hi}\perp\overline{gj}$ and $\angle i\cong\angle h$. complete the proof that $\overline{ij}\cong\overline{hj}$.
1 $\overline{hi}\perp\overline{gj}$
2 $\angle i\cong\angle h$
3 $\angle gjh\cong\angle gji$
4 $\overline{gj}\cong\overline{gj}$
5 $\triangle ghj\cong\triangle gij$
6 $\overline{ij}\cong\overline{hj}$

Explanation:

Step1: Definition of perpendicular lines

Since \( \overline{HI}\perp\overline{GJ}\), by the definition of perpendicular lines, \( \angle GJH = 90^{\circ}\) and \( \angle GJI=90^{\circ}\), so \( \angle GJH\cong\angle GJI\) (all right - angles are congruent).

Step2: Given information

\( \angle I\cong\angle H\) is given.

Step3: Reflexive property

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

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

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

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

Since \( \triangle GHJ\cong\triangle GIJ\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \overline{IJ}\cong\overline{HJ}\).

Answer:

1. Given (the problem statement says \( \overline{HI}\perp\overline{GJ}\))
2. Given (the problem statement says \( \angle I\cong\angle H\))
3. Definition of perpendicular lines (since \( \overline{HI}\perp\overline{GJ}\), \( \angle GJH = \angle GJI=90^{\circ}\))
4. Reflexive property of congruence (\( \overline{AB}\cong\overline{AB}\) for any segment \( \overline{AB}\))
5. AAS (Angle - Angle - Side) (\( \angle H\cong\angle I\), \( \angle GJH\cong\angle GJI\), \( \overline{GJ}\cong\overline{GJ}\))
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)