Question

complete the proof that ∠g ij ≅ ∠g ih.
(image of triangle with markings and a table with statements 1 - 5 and empty reason column)

Explanation:

Step1: Analyze Statement 1

The statement is \(\overline{GH} \cong \overline{GJ}\). From the diagram, we can see that \(GH\) and \(GJ\) are marked with the same number of tick marks, indicating they are congruent by the Given (or Definition of Congruent Segments from Diagram). So the reason for statement 1 is "Given (or Diagram shows congruent segments)".

Step2: Analyze Statement 2

The statement is \(\overline{HI} \cong \overline{IJ}\). Looking at the diagram, \(HI\) and \(IJ\) have the same tick marks, so this is also from the diagram (Given), so the reason is "Given (Diagram - congruent segments)".

Step3: Analyze Statement 3

The statement is \(\overline{GI} \cong \overline{GI}\). This is the Reflexive Property of Congruence (a segment is congruent to itself), so the reason is "Reflexive Property of Congruence".

Step4: Analyze Statement 4

The statement is \(\triangle GHI \cong \triangle GJI\). We have \( \overline{GH} \cong \overline{GJ}\), \( \overline{HI} \cong \overline{IJ}\), and \( \overline{GI} \cong \overline{GI}\), so by the SSS (Side - Side - Side) Congruence Postulate, \(\triangle GHI \cong \triangle GJI\). So the reason is "SSS Congruence Postulate".

Step5: Analyze Statement 5

The statement is \(\angle GJI \cong \angle GIH\). Since \(\triangle GHI \cong \triangle GJI\), corresponding parts of congruent triangles are congruent (CPCTC), so the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

1. Reason for \(\overline{GH} \cong \overline{GJ}\): Given (Diagram - congruent segments)
2. Reason for \(\overline{HI} \cong \overline{IJ}\): Given (Diagram - congruent segments)
3. Reason for \(\overline{GI} \cong \overline{GI}\): Reflexive Property of Congruence
4. Reason for \(\triangle GHI \cong \triangle GJI\): SSS Congruence Postulate
5. Reason for \(\angle GJI \cong \angle GIH\): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)