Sage has already proved that


I
G
J


F
G
H
. She is now attempting to prove that
¯¯¯¯¯¯
I
J

¯¯¯¯¯¯¯¯¯
F
H
. Help her to construct a viable argument. What is the missing statement and reason in her proof?

Statements Reasons
1.

I
G
J


F
G
H
1. Given
2. 2.
3.
¯¯¯¯¯¯
I
J

¯¯¯¯¯¯¯¯¯
F
H
3. If corresponding angles are congruent, then lines are parallel (1 point)
Responses

The missing statement is

G
I
J


G
F
H
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

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

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

J
I
F


G
F
H
and the missing reason is “CPCTC Theorem.”

1 answer

To prove that line segments \( \overline{IJ} \) and \( \overline{FH} \) are parallel based on the given triangles, we can use the fact that corresponding angles are congruent when two angles are formed by a transversal intersecting parallel lines.

Since Sage has already proven that triangles \( \triangle IGJ \) and \( \triangle FGH \) are similar, we know that their corresponding angles are congruent.

Thus, the missing statement and reason in Sage's proof are:

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

Hence the correct response is:

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