Question
What theorem proves that △ABC≅△DEF?
Option # 1: ASA
Option # 2: HL
Option # 3: SAS
Option # 4: SSA
Option # 1: ASA
Option # 2: HL
Option # 3: SAS
Option # 4: SSA
Answers
GPT-4o mini
To determine the correct theorem that proves that \( \triangle ABC \cong \triangle DEF \), we need to analyze the given options:
1. **ASA (Angle-Side-Angle)**: This theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
2. **HL (Hypotenuse-Leg)**: This theorem applies specifically to right triangles. If the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
3. **SAS (Side-Angle-Side)**: This theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
4. **SSA (Side-Side-Angle)**: This configuration does not guarantee congruence in general; it can lead to ambiguous results depending on the lengths of the sides involved.
Given the general information, if \( \triangle ABC \) is congruent to \( \triangle DEF \), it could be proven by either the ASA, SAS, or HL theorem, depending on the information known about angles and sides.
Without specific information about the angles and sides of the triangles, it is impossible to definitively select one theorem. However, SSA is not a suitable option for congruence because it does not guarantee that the triangles are congruent.
In conclusion, the correct answer would be either ASA, HL, or SAS, depending on the specific conditions. If you have more context on the sides and angles involved, I could help further narrow down the options.
1. **ASA (Angle-Side-Angle)**: This theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
2. **HL (Hypotenuse-Leg)**: This theorem applies specifically to right triangles. If the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
3. **SAS (Side-Angle-Side)**: This theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
4. **SSA (Side-Side-Angle)**: This configuration does not guarantee congruence in general; it can lead to ambiguous results depending on the lengths of the sides involved.
Given the general information, if \( \triangle ABC \) is congruent to \( \triangle DEF \), it could be proven by either the ASA, SAS, or HL theorem, depending on the information known about angles and sides.
Without specific information about the angles and sides of the triangles, it is impossible to definitively select one theorem. However, SSA is not a suitable option for congruence because it does not guarantee that the triangles are congruent.
In conclusion, the correct answer would be either ASA, HL, or SAS, depending on the specific conditions. If you have more context on the sides and angles involved, I could help further narrow down the options.