Question

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gj bisects ∠hgi. complete the proof that ij ≅ hj.

1 gj bisects ∠hgi
2 gi ≅ gh
3 ∠hgj ≅ ∠igj
4 gj ≅ gj
5 △ghj ≅ △gij
6 ij ≅ hj

Explanation:

Step1: Given

Given that \(\overrightarrow{GJ}\) bisects \(\angle HGI\).

Step2: Given

From the figure, \(\overline{GI}\cong\overline{GH}\) (marked as equal in the diagram).

Step3: Definition of angle - bisector

Since \(\overrightarrow{GJ}\) bisects \(\angle HGI\), by the definition of an angle - bisector, \(\angle HGJ\cong\angle IGJ\).

Step4: Reflexive property

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

Step5: 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. Here, in \(\triangle GHJ\) and \(\triangle GIJ\), \(\overline{GI}\cong\overline{GH}\), \(\angle HGJ\cong\angle IGJ\), \(\overline{GJ}\cong\overline{GJ}\), so \(\triangle GHJ\cong\triangle GIJ\) by SAS.

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

If \(\triangle GHJ\cong\triangle GIJ\), then their corresponding parts are congruent. So, \(\overline{IJ}\cong\overline{HJ}\) (because \(IJ\) and \(HJ\) are corresponding sides of congruent triangles \(\triangle GIJ\) and \(\triangle GHJ\)).

Answer:

1. Given
2. Given
3. Definition of angle - bisector
4. Reflexive property of congruence
5. SAS (Side - Angle - Side)
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)