Question

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gj bisects ∠hgi and (overline{gi}congoverline{gh}). complete the proof that (overline{ij}congoverline{hj}).

Explanation:

Step1: Definition of angle bisector

If a ray bisects an angle, it divides the angle into two congruent angles. So, if \( \overleftrightarrow{GJ}\) bisects \( \angle HGI\), then \( \angle HGJ\cong\angle IGJ\) (Reason for statement 3: Definition of angle - bisector).

Step2: Given information

Statement 2: \( \overline{GI}\cong\overline{GH}\) (Reason: Given).

Step3: Reflexive property

For any segment \( \overline{AB}\), \( \overline{AB}\cong\overline{AB}\). So, \( \overline{GJ}\cong\overline{GJ}\) (Reason for statement 4: Reflexive property of congruence).

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

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. In \( \triangle GHJ\) and \( \triangle GIJ\), we have \( \overline{GI}\cong\overline{GH}\) (side), \( \angle HGJ\cong\angle IGJ\) (angle), \( \overline{GJ}\cong\overline{GJ}\) (side). So, \( \triangle GHJ\cong\triangle GIJ\) (Reason for statement 5: SAS congruence criterion).

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

If \( \triangle GHJ\cong\triangle GIJ\), then their corresponding parts are congruent. So, \( \overline{IJ}\cong\overline{HJ}\) (Reason for statement 6: CPCTC).

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Definition of angle - bisector
4. Reason: Reflexive property of congruence
5. Reason: SAS congruence criterion
6. Reason: CPCTC