Sage has already proved that △IGJ∼△FGH

Question

. 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.”

Answer 1

To prove that $IJ\parallel FH$ using the fact that $\mathrm{△}IGJ\sim \mathrm{△}FGH$, we need to establish that corresponding angles in the two triangles are congruent. The statement that corresponds to this relationship would relate the angles located at the vertices of each triangle that correspond to each other.

Given that $\mathrm{△}IGJ\sim \mathrm{△}FGH$, one pair of corresponding angles are $\mathrm{\angle }GJI$ and $\mathrm{\angle }GHF$.

So, to fill in your table:

1. Statements

   • $\mathrm{△}IGJ\sim \mathrm{△}FGH$
   • $\mathrm{\angle }GJI\cong \mathrm{\angle }GHF$ (missing statement)
   • $IJ\parallel FH$

2. Reasons

   • Given
   • Corresponding angles of similar triangles are congruent. (missing reason)
   • If corresponding angles are congruent, then lines are parallel.

Thus, the complete argument would be:

• The missing statement is $\mathrm{\angle }GJI\cong \mathrm{\angle }GHF$.
• The missing reason is "corresponding angles of similar triangles are congruent."

This supports the conclusion that $IJ\parallel FH$ based on the properties of the angles in similar triangles.