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Sage has already proved that △IGJ∼△FGH . She is now attempting to prove that IJ¯¯¯¯¯¯∥FH¯¯¯¯¯¯¯¯ . Help her to construct a viable argument. What is the missing statement and reason in her proof?

Statements Reasons
1. △IGJ∼△FGH 1. Given
2. 2.
3. IJ¯¯¯¯¯¯∥FH¯¯¯¯¯¯¯¯ 3. If corresponding angles are congruent, then lines are parallel
(1 point)
Responses

The missing statement is ∠JIF≅∠GFH and the missing reason is “CPCTC Theorem.”
The missing statement is angle upper J upper I upper F congruent to angle upper G upper F upper H and the missing reason is “CPCTC Theorem.”

The missing statement is ∠JIF≅∠GFH and the missing reason is “corresponding angles of similar triangles are congruent.”
The missing statement is angle upper J upper I upper F congruent to angle upper G upper F upper H and the missing reason is “corresponding angles of similar triangles are congruent.”

The missing statement is ∠GJI≅∠GHF and the missing reason is “corresponding angles of similar triangles are congruent.”
The missing statement is angle upper G upper J upper I congruent to angle upper G upper H upper F and the missing reason is “corresponding angles of similar triangles are congruent.”

The missing statement is ∠GIJ≅∠GFH and the missing reason is “CPCTC Theorem.”
The missing statement is angle upper G upper I upper J congruent to angle upper G upper F upper H and the missing reason is “CPCTC Theorem.”

1 answer

To establish the argument that if triangles \( \triangle IGJ \sim \triangle FGH \), then lines \( \overline{IJ} \parallel \overline{FH} \), we need a correspondence of angles. Since similarity of triangles implies that corresponding angles are equal, we can assert this property in our proof.

Given the similarity \( \triangle IGJ \sim \triangle FGH \):

  1. The corresponding angles in the similar triangles are congruent, which leads to: \[ \angle JIF \cong \angle GFH. \] or, depending on the corresponding labels, \( \angle GJI \cong \angle GHF \).

  2. With the corresponding angles established as congruent, we can use the relationship between angles and parallel lines: \[ IJ \parallel FH \text{ (if the corresponding angles are congruent)}. \]

Thus, the missing statement is either:

  • \( \angle JIF \cong \angle GFH \) or
  • \( \angle GJI \cong \angle GHF \)

And both statements can justify the reasoning as: "corresponding angles of similar triangles are congruent."

Here is how the complete argument looks:

Statements:

  1. \( \triangle IGJ \sim \triangle FGH \)
  2. \( \angle JIF \cong \angle GFH \) or any of the corresponding angle pairs
  3. \( IJ \parallel FH \)

Reasons:

  1. Given
  2. Corresponding angles of similar triangles are congruent.
  3. If corresponding angles are congruent, then lines are parallel.

Given these options, the suitable choice based on the arguments would be: The missing statement is \( \angle JIF \cong \angle GFH \) and the missing reason is “corresponding angles of similar triangles are congruent.”