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 “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 ∠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 ∠GJI≅∠GHF and the missing reason is “corresponding angles of similar triangles are congruent.”

Answer 1

To help Sage construct a viable argument proving that \( IJ \parallel FH \) given that \( \triangle IGJ \sim \triangle FGH \), we need to identify the appropriate missing statement and reason.

Since \( \triangle IGJ \sim \triangle FGH \), we know that the corresponding angles of the two triangles are congruent. To show that the lines \( IJ \) and \( FH \) are parallel, we can use the fact that if a pair of corresponding angles are congruent, then the lines are parallel.

The statement will involve the angles at the vertices where the lines \( IJ \) and \( FH \) are located.

Thus, the correct missing statement should be that the corresponding angles are congruent, such as:

• Missing Statement: \( \angle JIF \cong \angle GFH \) (this indicates one pair of corresponding angles.)

And the reasoning that accompanies this should be:

• Missing Reason: "Corresponding angles of similar triangles are congruent."

So the correct argument would look like this:

1. \( \triangle IGJ \sim \triangle FGH \) (Given)
2. \( \angle JIF \cong \angle GFH \) (Corresponding angles of similar triangles are congruent.)
3. \( IJ \parallel FH \) (If corresponding angles are congruent, then lines are parallel.)

Among your provided options, the correct choice is: The missing statement is \( \angle JIF \cong \angle GFH \) and the missing reason is "corresponding angles of similar triangles are congruent."