To show that triangles \( \triangle ABC \cong \triangle EFD \) by mapping \( \triangle ABC \) on top of \( \triangle EFD \), we typically need to establish congruence through specific transformations or congruence criteria. The most common transformations include translations, rotations, and reflections.
- Transformations Needed: We should check the corresponding sides and angles to determine which congruence criteria can be applied. The transformations will depend on the specific coordinates or measurements of the angles and sides of the triangles.
Based on the congruence criteria listed, the following responses could apply (assuming appropriate configurations in the triangles):
- AAS (Angle-Angle-Side): If two angles and the non-included side of triangle \(ABC\) correspond to two angles and the non-included side of triangle \(EFD\), this criterion would apply.
- ASA (Angle-Side-Angle): Similarly, if two angles and the included side correspond, ASA applies.
- SSS (Side-Side-Side): If all three sides of triangle \(ABC\) correspond to all three sides of triangle \(EFD\), we can use SSS.
Note: The criteria SAS, AAS, or ASA could also suffice, depending on which sides and angles are known to be congruent.
- Theorem Justification: The theorem that is used to justify the congruence depends on the specific pairs of angles and sides being compared:
- If using AAS, it would justify that two angles and a corresponding non-included side are known, showing congruence.
- If using ASA, it would show two angles and the included side are known.
- If using SSS, it would confirm all three sides are equal.
- HL (Hypotenuse-Leg) only applies to right triangles.
Given the options presented, I would suggest confirming the specific angles and sides you are looking at in triangles \(ABC\) and \(EFD\) to choose correctly among AAS, SAS, ASA, or SSS based on what is given.
In summary: The specific transformations needed depend on the measurements. Possible congruence theorems include AAS, ASA, and SSS depending on the situation.