To determine the additional congruence statement based on the diagram and the information given, we need to analyze the statements provided:
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Given Information:
- \( \angle HGI = \angle JGI \)
- \( \angle GIH = \angle GJI \)
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Congruence Options:
- Option #1: \( \angle H = \angle J \)
- Option #2: \( GI = GI \) (Reflexive property)
- Option #3: \( HI = I \) (This option seems incorrectly stated)
- Option #4: \( GH = GJ \)
Based on the given angles being congruent, we can conclude:
- From the angles given, we have two pairs of angles congruent:
- \( \angle HGI = \angle JGI \)
- \( \angle GIH = \angle GJI \)
Assuming that the sides \( GH \) and \( GJ \) are also sides of the triangles involving points \( H \) and \( J \) respectively, we can utilize that information to arrive at congruence.
Now, since we have:
- 1 pair of angles (from \( \angle HGI \) and \( \angle JGI \))
- 1 pair of angles (from \( \angle GIH \) and \( \angle GJI \))
- The side \( GI \) which is common (reflexive property)
This suggests that the two triangles \( AGHI \) and \( AGJI \) can be compared using the Angle-Angle-Side (AAS) congruence criterion.
Conclusion:
- Additional Congruence Statement: Based on the analysis, the statement \( GI = GI \) (Option #2) can always be said, but it does not provide new information.
- For triangle congruence, the best choice becomes:
- Option #3 (AAS) would verify the triangles by using the two sets of angles plus the included side.
Therefore, for your second question, the option that proves triangle congruence \( AGHI \cong AGJI \) is:
Option #3: AAS
To summarize:
- Additional Congruence Statement: Option #2: \( GI = GI \) (but it's not completely informative).
- Triangle Congruence Proof: Option #3: AAS.