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 ∠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.” Skip to navigation

Answer 1

To prove that \( IJ \parallel FH \) given that \( \triangle IGJ \sim \triangle FGH \), we can utilize the properties of similar triangles, specifically the congruence of corresponding angles.

Since the triangles are similar, we know that:

1. The angles of \( \triangle IGJ \) are proportional to the angles of \( \triangle FGH \).
2. Therefore, the corresponding angles are congruent.

Since you already have statement 1 that \( \triangle IGJ \sim \triangle FGH \), the next logical step is to state which pairs of angles correspond and conclude from that their congruence.

A plausible choice for the missing statement and reason could be:

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

Thus, the argument built from this statement leads to the conclusion that if two lines have corresponding angles that are congruent, the lines are parallel, which directly supports the claim \( IJ \parallel FH \).

Here's how to fit it all together in your proof:

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)

Based on the options you provided, the correct choice regarding the missing statement and reason seems to be:

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