Question
Is △ABC≅△DFE by AAS Theorem? (1 point) Responses No. The ASA Theorem is used.
Yes. The AAS Theorem is used.
Yes. BC¯¯¯¯¯¯¯¯≅FE¯¯¯¯¯¯¯¯.
No. DE¯¯¯¯¯¯¯¯ does not correspond to BC¯¯¯¯¯¯¯¯.
Yes. The AAS Theorem is used.
Yes. BC¯¯¯¯¯¯¯¯≅FE¯¯¯¯¯¯¯¯.
No. DE¯¯¯¯¯¯¯¯ does not correspond to BC¯¯¯¯¯¯¯¯.
Answers
GPT-4o mini
To determine whether triangles \( \triangle ABC \) and \( \triangle DFE \) are congruent by the AAS (Angle-Angle-Side) Theorem, we need to check if two angles and the included side of one triangle correspond to two angles and the included side of the other triangle.
1. **Identify Angles and Sides**: If we know that:
- \( \angle A \equiv \angle D \) (corresponding angles),
- \( \angle B \equiv \angle F \) (corresponding angles),
- \( BC \equiv FE \) (corresponding sides),
then we can say that \( \triangle ABC \cong \triangle DFE \) by the AAS theorem.
2. **Responses Evaluation**:
- If the statement says "No. The ASA Theorem is used," it suggests using Angle-Side-Angle instead of AAS. This is not correct if we are indeed using the AAS theorem properly as described.
- The response "Yes. The AAS Theorem is used" would be correct if the conditions for AAS are met based on angles and sides.
- The response "Yes. \( BC \equiv FE \)" can be a key detail in supporting congruence if angles were also proven to be congruent.
- The last response "No. \( DE \) does not correspond to \( BC \)" implies a potential mismatch in parts of triangles which could invalidate congruence by AAS.
To conclude, the correct response depends on whether the angles and sides align as required by AAS. If the angles and sides are appropriately matched, then "Yes. The AAS Theorem is used" would be the correct answer. If there is any ambiguity or mismatch, then the last option might apply.
In summary, if you have two angles and a non-included side between them corresponding in both triangles, then **the answer is**:
**Yes. The AAS Theorem is used.**
1. **Identify Angles and Sides**: If we know that:
- \( \angle A \equiv \angle D \) (corresponding angles),
- \( \angle B \equiv \angle F \) (corresponding angles),
- \( BC \equiv FE \) (corresponding sides),
then we can say that \( \triangle ABC \cong \triangle DFE \) by the AAS theorem.
2. **Responses Evaluation**:
- If the statement says "No. The ASA Theorem is used," it suggests using Angle-Side-Angle instead of AAS. This is not correct if we are indeed using the AAS theorem properly as described.
- The response "Yes. The AAS Theorem is used" would be correct if the conditions for AAS are met based on angles and sides.
- The response "Yes. \( BC \equiv FE \)" can be a key detail in supporting congruence if angles were also proven to be congruent.
- The last response "No. \( DE \) does not correspond to \( BC \)" implies a potential mismatch in parts of triangles which could invalidate congruence by AAS.
To conclude, the correct response depends on whether the angles and sides align as required by AAS. If the angles and sides are appropriately matched, then "Yes. The AAS Theorem is used" would be the correct answer. If there is any ambiguity or mismatch, then the last option might apply.
In summary, if you have two angles and a non-included side between them corresponding in both triangles, then **the answer is**:
**Yes. The AAS Theorem is used.**