H

Question

H
J
F
I
G 
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 ∠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 “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.”

GPT-4o mini · Answer

In order to prove that $ IJ \parallel FH $ based on the similarity of triangles $ \triangle IGJ \sim \triangle FGH $, we need to demonstrate that the corresponding angles of the similar triangles are congruent. The correct choice for the missing statement and reason is: **Missing Statement:** $ \angle GJI \cong \angle GHF $ **Missing Reason:** “corresponding angles of similar triangles are congruent.” This works because if the triangles are similar, their corresponding angles must be equal, which is a foundational property of similar triangles. Then, since the angles are congruent, we can conclude that the lines $ IJ $ and $ FH $ are parallel by the Converse of the Corresponding Angles Postulate. So, the complete proof would look like this: **Statements** | **Reasons** --- | --- 1. $ \triangle IGJ \sim \triangle FGH $ | Given 2. $ \angle GJI \cong \angle GHF $ | Corresponding angles of similar triangles are congruent 3. $ IJ \parallel FH $ | If corresponding angles are congruent, then lines are parallel This establishes the necessary arguments for Sage’s proof.