△ABC≅△ABD

Question

Using the info provided in the picture below, which theorem proves this?

(1 point)
Responses which answer is correct and why

SAS Theorem
SAS Theorem

HL Theorem
HL Theorem

SSS Theorem
SSS Theorem

ASA Theorem
ASA Theorem
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Answer 1

To determine which theorem proves that triangle \( \triangle ABC \) is congruent to triangle \( \triangle ABD \), we need to analyze the conditions typically required by the different congruence theorems:

1. SAS (Side-Angle-Side) Theorem: Two triangles are congruent if two sides and the included angle of one triangle are equal to those of another triangle.

2. HL (Hypotenuse-Leg) Theorem: This applies specifically to right triangles, stating that 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. SSS (Side-Side-Side) Theorem: Two triangles are congruent if all three sides of one triangle are equal to the three sides of another triangle.

4. ASA (Angle-Side-Angle) Theorem: Two triangles are congruent if two angles and the included side of one triangle are equal to those of another triangle.

Without specific information regarding the sides and angles of triangles \( ABC \) and \( ABD \), it is difficult to determine which theorem applies. However, based on the names of the triangles and the common conditions:

If \( \triangle ABC \) and \( \triangle ABD \) share one side \( AB \) and you have equal sides \( AC \) and \( AD \), or if you know that an angle is included, you might be able to use SAS or ASA.

If you have the lengths of all three sides (like \( AB, AC, AD \)), then SSS would be applicable.

In the absence of the visuals and specific details from the given "picture below", one possible and likely theorem could be SAS Theorem since it usually involves triangles that share a side and related angles. However, a better answer could be derived if more details about the sides and angles of the triangles were provided.

In conclusion, based on the typical relationships in triangle comparison:

• If you have two sides and the included angle equal, use SAS.
• If you have two angles and the included side equal, use ASA.
• If you have all three sides equal, use SSS.
• If both triangles are right angled, and legs or hypotenuses are known, you may consider HL.

So, the answer would depend entirely on the specific details provided in the picture not given in the question. Based on a typical scenario, I recommend answering SAS Theorem if two sides and the angle between are given, but please verify with the actual picture data you have.