Bot GPT 4-o mini To prove that triangles FHG and KJG are congruent by the ASA (Angle-Side-Angle) postulate, we need to identify pairs of angles and the included side that are congruent.
Given that \( HF \parallel JK \) and \( HG \cong JG \), we can use the following reasoning:
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Angles: Since \( HF \parallel JK \) and \( HG \) and \( JG \) are transversals intersecting these parallel lines, the corresponding angles formed are congruent.
Specifically, we can use:
- \( \angle HFG \) (formed by \( HF \) and \( HG \)) and \( \angle KJG \) (formed by \( JK \) and \( JG \)) are alternate interior angles. Since \( HF \) and \( JK \) are parallel, we have:
- \( \angle HFG \cong \angle KJG \) (by the Corresponding Angles Postulate).
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Included Side: We know:
- \( HG \cong JG \) (given).
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Final Angle: The other angle in each triangle can be identified:
- \( \angle FHG \) and \( \angle KJG \) are the same angle as they both share angle \( \angle G \) with the exterior angles being their pair.
Thus, we establish:
- \( \angle HFG \cong \angle KJG \) (as discussed)
- \( HG \cong JG \) (given)
- Having the corresponding angles from parallel lines, we can say \( \angle FHG \cong \angle KJG \).
Therefore, using the information:
- Statement: \( \angle HFG \cong \angle KJG \) because alternate interior angles are congruent.
This would be one of the statements needed to establish the congruence of triangles \( FHG \) and \( KJG \) using the ASA postulate.
Hence, the correct statement is:
Angle HFG ≅ Angle KJG because alternate interior angles are congruent.
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