Question

complete the proof that ∠fhg ≅ ∠fhi.
image of a geometric figure with triangles and congruency marks, and a table with statements 1 - 5 and empty reason columns:

1. ( overline{fi} cong overline{fg} )
2. ( overline{hi} cong overline{gh} )
3. ( overline{fh} cong overline{fh} )
4. ( \triangle fgh cong \triangle fih )
5. ( angle fhg cong angle fhi )

Explanation:

Step1: Identify Given Congruences

From the diagram, \( \overline{FI} \cong \overline{FG} \) (marked with two ticks), \( \overline{HI} \cong \overline{GH} \) (marked with one tick), and \( \overline{FH} \cong \overline{FH} \) (common side).

Step2: Apply SSS Congruence

For triangles \( \triangle FGH \) and \( \triangle FIH \), we have:

• \( \overline{FI} \cong \overline{FG} \) (Given)
• \( \overline{HI} \cong \overline{GH} \) (Given)
• \( \overline{FH} \cong \overline{FH} \) (Reflexive Property of Congruence)

By the Side - Side - Side (SSS) Congruence Postulate, \( \triangle FGH \cong \triangle FIH \).

Step3: Corresponding Angles Congruence

Since \( \triangle FGH \cong \triangle FIH \), their corresponding angles \( \angle FHG \) and \( \angle FHI \) are congruent (Corresponding Parts of Congruent Triangles are Congruent, CPCTC).

Now, let's fill in the reasons:

1. \( \overline{FI} \cong \overline{FG} \): Given (from the diagram markings)
2. \( \overline{HI} \cong \overline{GH} \): Given (from the diagram markings)
3. \( \overline{FH} \cong \overline{FH} \): Reflexive Property of Congruence (a segment is congruent to itself)
4. \( \triangle FGH \cong \triangle FIH \): SSS Congruence Postulate (since three sides of \( \triangle FGH \) are congruent to three sides of \( \triangle FIH \))
5. \( \angle FHG \cong \angle FHI \): CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Reflexive Property of Congruence
4. Reason: SSS Congruence Postulate
5. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)