Question
Sage has already proved that \(\triangle IGJ\sim\triangle FGH\). She is now attempting to prove that \(\overline{IJ}\parallel\overline{FH}\). Help her to construct a viable argument. What is the missing statement and reason in her proof? Statements Reasons 1. \(\triangle IGJ\sim\triangle FGH\) 1. Given 2. 2. 3. \(\overline{IJ}\parallel\overline{FH}\) 3. If corresponding angles are congruent, then lines are parallel (1 point) Responses The missing statement is \(\angle GIJ\cong\angle GFH\) 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 \(\angle GJI\cong\angle GHF\) 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 \(\angle JIF\cong\angle GFH\) 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 \(\angle JIF\cong\angle GFH\) 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
Answers
GPT-4o mini
To prove that \(\overline{IJ} \parallel \overline{FH}\), Sage can use the fact that corresponding angles are congruent in similar triangles. Since she has established that \(\triangle IGJ \sim \triangle FGH\), it follows that their corresponding angles are congruent.
In this case:
- The corresponding angles are:
- \(\angle GIJ\) (from \(\triangle IGJ\)) and \(\angle GFH\) (from \(\triangle FGH\)).
Thus, the missing statement should be:
**Missing Statement:** \(\angle GIJ \cong \angle GFH\)
Now, the reason supporting this conclusion is based on the properties of similar triangles. Since the triangles are similar, their corresponding angles are congruent:
**Missing Reason:** "Corresponding angles of similar triangles are congruent."
Consequently, since \(\angle GIJ \cong \angle GFH\), by the converse of the Corresponding Angles Postulate, we can conclude that:
\(\overline{IJ} \parallel \overline{FH}\).
So, Sage's argument can be summarized as follows:
1. \(\triangle IGJ \sim \triangle FGH\) - Given
2. \(\angle GIJ \cong \angle GFH\) - Corresponding angles of similar triangles are congruent
3. \(\overline{IJ} \parallel \overline{FH}\) - If corresponding angles are congruent, then lines are parallel.
Therefore, the correct response would be that the missing statement is \(\angle GIJ \cong \angle GFH\) and the missing reason is "corresponding angles of similar triangles are congruent."
In this case:
- The corresponding angles are:
- \(\angle GIJ\) (from \(\triangle IGJ\)) and \(\angle GFH\) (from \(\triangle FGH\)).
Thus, the missing statement should be:
**Missing Statement:** \(\angle GIJ \cong \angle GFH\)
Now, the reason supporting this conclusion is based on the properties of similar triangles. Since the triangles are similar, their corresponding angles are congruent:
**Missing Reason:** "Corresponding angles of similar triangles are congruent."
Consequently, since \(\angle GIJ \cong \angle GFH\), by the converse of the Corresponding Angles Postulate, we can conclude that:
\(\overline{IJ} \parallel \overline{FH}\).
So, Sage's argument can be summarized as follows:
1. \(\triangle IGJ \sim \triangle FGH\) - Given
2. \(\angle GIJ \cong \angle GFH\) - Corresponding angles of similar triangles are congruent
3. \(\overline{IJ} \parallel \overline{FH}\) - If corresponding angles are congruent, then lines are parallel.
Therefore, the correct response would be that the missing statement is \(\angle GIJ \cong \angle GFH\) and the missing reason is "corresponding angles of similar triangles are congruent."