Question
ABC and DEF are two congruent triangles, in which BAC = EDF, AB = DE & AC = DF.
Which of the conditions given below is valid?
Select an answer:
RHS
AAS
ASA
SSS
SAS
Which of the conditions given below is valid?
Select an answer:
RHS
AAS
ASA
SSS
SAS
Answers
GPT-4o mini
To determine which condition is valid for proving that triangles ABC and DEF are congruent, let's analyze the given information:
1. **BAC = EDF**: This indicates that the angles are equal (Angle).
2. **AB = DE**: This indicates that one side of the triangle is equal to the corresponding side of the other triangle (Side).
3. **AC = DF**: This indicates that another side of the triangle is equal to another corresponding side of the other triangle (Side).
From the information provided, we can apply the following congruence criteria:
- **ASA (Angle-Side-Angle)**: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. We have one angle (BAC = EDF) and the two sides (AB = DE and AC = DF), satisfying the ASA condition.
- **AAS (Angle-Angle-Side)**: If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent. We can potentially use AAS as well, since we could find another angle in both triangles (say angle C is equal to angle F if we look at the overall triangle).
- **SAS (Side-Angle-Side)**: From the information we have, we could also satisfy SAS if we consider the side AB, the angle BAC, and the side AC.
The more straightforward approach here is to recognize ASA or SAS as valid due to the angle and the two sides, but since we don’t have an explicit included angle for the sides provided, ASA is more aligned with the information.
Therefore, the valid condition from the options listed is:
**ASA**.
1. **BAC = EDF**: This indicates that the angles are equal (Angle).
2. **AB = DE**: This indicates that one side of the triangle is equal to the corresponding side of the other triangle (Side).
3. **AC = DF**: This indicates that another side of the triangle is equal to another corresponding side of the other triangle (Side).
From the information provided, we can apply the following congruence criteria:
- **ASA (Angle-Side-Angle)**: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. We have one angle (BAC = EDF) and the two sides (AB = DE and AC = DF), satisfying the ASA condition.
- **AAS (Angle-Angle-Side)**: If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent. We can potentially use AAS as well, since we could find another angle in both triangles (say angle C is equal to angle F if we look at the overall triangle).
- **SAS (Side-Angle-Side)**: From the information we have, we could also satisfy SAS if we consider the side AB, the angle BAC, and the side AC.
The more straightforward approach here is to recognize ASA or SAS as valid due to the angle and the two sides, but since we don’t have an explicit included angle for the sides provided, ASA is more aligned with the information.
Therefore, the valid condition from the options listed is:
**ASA**.