Question

bisects ∠hg and gt = gh. complete the proof that h = h.

1 bisects ∠hgj
2 gt = gh
3 ∠hgj ≅ ∠igj
4 gj ≅ gj
5 △ghj ≅ △gji
6 hj ≅ hj

Explanation:

Step1: Given

The first statement is given in the problem.

Step2: Given

The second statement is given in the problem.

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 of congruence

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

Step5: SAS (Side - Angle - Side) congruence criterion

We have \(\overline{GH}\cong\overline{GI}\) (given), \(\angle HGJ\cong\angle IGJ\) (from step 3), and \(\overline{GJ}\cong\overline{GJ}\) (from step 4). So, by the SAS (Side - Angle - Side) congruence criterion, \(\triangle GHJ\cong\triangle GIJ\).

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

Since \(\triangle GHJ\cong\triangle GIJ\), then their corresponding parts \(\overline{HJ}\) and \(\overline{IJ}\) are congruent. So \(\overline{HJ}\cong\overline{IJ}\) by CPCTC.

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)