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 ∠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 ∠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 ∠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.” 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.” Skip to navigation

Answer 1

To construct a viable argument in proving that \( IJ \parallel FH \) given that \( \triangle IGJ \sim \triangle FGH \), we can use the property of corresponding angles in similar triangles.

1. Since \( \triangle IGJ \sim \triangle FGH \), we know the corresponding angles are congruent.
2. Therefore, we can state that either:
   • \( \angle GJI \cong \angle GHF \) or
   • \( \angle JIF \cong \angle GFH \)

To support the conclusion that the lines \( IJ \) and \( FH \) are parallel, we need the angles that are formed by the transversal at points \( I \) and \( F \) and the corresponding angles at the other triangle’s points.

The missing statement can therefore be:

• The missing statement is \( \angle GJI \cong \angle GHF \)
• The missing reason is "corresponding angles of similar triangles are congruent."

This leads to the conclusion that if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel, thus proving \( IJ \parallel FH \).

Thus, the correct choice would be:

The missing statement is \( \angle GJI \cong \angle GHF \) and the missing reason is “corresponding angles of similar triangles are congruent.”