Sage has already proved that triangle IGJ ~ triangle 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?

Question

Sage has already proved that triangle IGJ ~ triangle 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. Triangle IGJ ~ triangle FGH. 1. Given.

2. Blank. 2. Blank.

3. IJ||FH. 3. If corresponding angles are congruent, then lines are parallel.
-----------------------------------------------------------------------------------------------
A. The missing statement is <JIF is congruent to <GFH and the missing reason is "CPCTC Theorem."
B. The missing statement is <GJI is congruent to <GHF and the missing reason is "corresponding angles of similar triangles are congruent."
C. The missing statement is <JIF is congruent to <GFH and the missing reacon is "corresponding angles of similar triangles are congruent."
D. The missing statement is <GIJ is congruent to <GFH and the missing reacon is "CPCTC Theorem."

Answer 1

Bot GPT 4-o mini answered
2 hours ago

To prove that \( IJ \parallel FH \) given that triangle \( IGJ \sim \) triangle \( FGH \), we know that corresponding angles in similar triangles are congruent. Since \( IJ \parallel FH \) is based on the fact that corresponding angles are congruent, we need to identify the correct angles and the reasoning behind this congruence.

In this scenario, the key angles of interest are any angles formed by the lines \( IJ \) and \( FH \) when intersected by a transversal (like line \( GH \)).

Looking at the options provided, the best choice aligns the correct pair of angles and the correct reasoning based on the properties of similar triangles. The missing statement should point to one of the corresponding angles between the two triangles that are congruent.

B. The missing statement is \( \angle GJI \cong \angle GHF \) and the missing reason is "corresponding angles of similar triangles are congruent."

This option correctly identifies two angles in triangles \( IGJ \) and \( FGH \) that are corresponding angles (thus must be congruent), supporting the conclusion that the lines \( IJ \) and \( FH \) are parallel.