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Question

A triangle is labeled clockwise as upper F upper G upper H, with vertex F pointing left. Point upper I is marked at the midpoint of side upper F upper G. Point upper J is marked at the midpoint of side upper G upper H. A line connects upper I and upper J.

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 ∠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 ∠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.”
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Answer 1

To help Sage prove that \( IJ \parallel FH \), we need to identify the missing statement and reason in her proof. Given that the triangles are similar (\( \triangle IGJ \sim \triangle FGH \)), we can conclude that corresponding angles from these triangles are also congruent.

The correct missing statement and reason should focus on the corresponding angles between the two similar triangles.

Missing Statement: \( \angle JIF \cong \angle GFH \)
Missing Reason: "Corresponding angles of similar triangles are congruent."

Thus, this option correctly identifies the corresponding angles that lead to the conclusion that lines are parallel.

So the answer is:

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