Question

gh ≅ gj and hi ≅ ij. complete the proof that ∠h ≅ ∠j.

statement | reason
--- | ---
1 ( overline{gh} cong overline{gj} ) |
2 ( overline{hi} cong overline{ij} ) |
3 ( overline{gi} cong overline{gi} ) |
4 ( \triangle ghi cong \triangle gji ) |
5 ( angle h cong angle j ) |

Explanation:

Step1: Identify Given Information

We know \(\overline{GH} \cong \overline{GJ}\) (given), \(\overline{HI} \cong \overline{IJ}\) (given), and \(\overline{GI} \cong \overline{GI}\) (reflexive property of congruence).

Step2: Prove Triangle Congruence

By the Side - Side - Side (SSS) congruence criterion, since \(\overline{GH} \cong \overline{GJ}\), \(\overline{HI} \cong \overline{IJ}\), and \(\overline{GI} \cong \overline{GI}\), we can conclude that \(\triangle GHI \cong \triangle GJI\).

Step3: Corresponding Parts of Congruent Triangles

Since \(\triangle GHI \cong \triangle GJI\), their corresponding angles are congruent. The angle \(\angle H\) in \(\triangle GHI\) corresponds to angle \(\angle J\) in \(\triangle GJI\), so \(\angle H \cong \angle J\) (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Now let's fill in the reasons for each statement:

• Statement 1: \(\overline{GH} \cong \overline{GJ}\), Reason: Given
• Statement 2: \(\overline{HI} \cong \overline{IJ}\), Reason: Given
• Statement 3: \(\overline{GI} \cong \overline{GI}\), Reason: Reflexive property of congruence
• Statement 4: \(\triangle GHI \cong \triangle GJI\), Reason: SSS (Side - Side - Side) congruence criterion
• Statement 5: \(\angle H \cong \angle J\), Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

To complete the proof:

1. Reason for \(\overline{GH} \cong \overline{GJ}\): Given
2. Reason for \(\overline{HI} \cong \overline{IJ}\): Given
3. Reason for \(\overline{GI} \cong \overline{GI}\): Reflexive property of congruence
4. Reason for \(\triangle GHI \cong \triangle GJI\): SSS (Side - Side - Side) congruence criterion
5. Reason for \(\angle H \cong \angle J\): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)