Question

Sage has already proved that △IGJ∼△FGH
△
𝐼
𝐺
𝐽
∼
△
𝐹
𝐺
𝐻
. She is now attempting to prove that IJ¯¯¯¯¯¯∥FH¯¯¯¯¯¯¯¯
𝐼
𝐽
¯
∥
𝐹
𝐻
¯
. Help her to construct a viable argument. What is the missing statement and reason in her proof?

Complete the Table to Prove Two Given Lines Are Parallel
Statements Reasons
1. △IGJ∼△FGH
△
𝐼
𝐺
𝐽
∼
△
𝐹
𝐺
𝐻
1. Given
2. 2.
3. IJ¯¯¯¯¯¯∥FH¯¯¯¯¯¯¯¯
𝐼
𝐽
¯
∥
𝐹
𝐻
¯
3. If corresponding angles are congruent, then lines are parallel

The missing statement is ∠GJI≅∠GHF ∠ 𝐺 𝐽 𝐼 ≅ ∠ 𝐺 𝐻 𝐹 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 ∠ 𝐽 𝐼 𝐹 ≅ ∠ 𝐺 𝐹 𝐻 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 ∠ 𝐽 𝐼 𝐹 ≅ ∠ 𝐺 𝐹 𝐻 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 ∠

Answer 1

Missing statement (line 2): ∠GJI ≅ ∠GHF

Missing reason (line 2): Corresponding angles of similar triangles are congruent.

So the completed proof is:
1. △IGJ ∼ △FGH — Given
2. ∠GJI ≅ ∠GHF — Corresponding angles of similar triangles are congruent
3. IJ ∥ FH — If corresponding angles are congruent, then lines are parallel