Question

complete the proof that ∠h ≅ ∠j.
statement 1: overline{hi} ≅ overline{ij}
reason: blank
statement 2: overline{gj} ≅ overline{gh}
reason: blank
statement 3: overline{gi} ≅ overline{gi}
reason: blank
statement 4: △ghi ≅ △gji
reason: blank
statement 5: ∠h ≅ ∠j
reason: blank

Explanation:

Step1: Analyze Statement 1

The statement is \(\overline{HI} \cong \overline{IJ}\). From the diagram, we can see that the segments \(HI\) and \(IJ\) have the same number of tick marks, which indicates they are congruent by the definition of congruent segments (marked congruent segments are congruent). So the reason for \(\overline{HI} \cong \overline{IJ}\) is "Given (or Marked Congruent Segments)".

Step2: Analyze Statement 2

The statement is \(\overline{GJ} \cong \overline{GH}\). Looking at the diagram, \(GJ\) and \(GH\) have the same number of hash marks, so they are congruent by the definition of congruent segments (marked congruent segments are congruent). So the reason for \(\overline{GJ} \cong \overline{GH}\) is "Given (or Marked Congruent Segments)".

Step3: Analyze Statement 3

The statement is \(\overline{GI} \cong \overline{GI}\). This is the reflexive property of congruence, which states that any 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 \(\overline{HI} \cong \overline{IJ}\) (from step 1), \(\overline{GH} \cong \overline{GJ}\) (from step 2), and \(\overline{GI} \cong \overline{GI}\) (from step 3). By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So the reason is "SSS Congruence Postulate".

Step5: Analyze Statement 5

The statement is \(\angle H \cong \angle J\). Since \(\triangle GHI \cong \triangle GJI\) (from step 4), corresponding parts of congruent triangles are congruent (CPCTC). So the reason is "CPCTC (Corresponding Parts of Congruent Triangles are Congruent)".

Answer:

For Statement 1: Reason is "Marked Congruent Segments (or Given)"; For Statement 2: Reason is "Marked Congruent Segments (or Given)"; For Statement 3: Reason is "Reflexive Property of Congruence"; For Statement 4: Reason is "SSS Congruence Postulate"; For Statement 5: Reason is "CPCTC"