To show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle EFD \) (notated as \( \triangle ABC \cong \triangle EFD \)), we need to identify the transformations that can map triangle \( ABC \) onto triangle \( EFD \). The transformations that maintain congruence are:
- Reflection: If \( EFD \) is a mirror image of \( ABC \), a reflection can map the two triangles onto each other.
- Rotation: If \( ABC \) needs to be turned to align with \( EFD \), a rotation can achieve this.
- Translation: If \( ABC \) simply needs to be slid over to match \( EFD \), then a translation is required.
A dilation, however, changes the size of the shape, which would not show congruence. Therefore, the transformations that apply are:
- Reflection
- Rotation
- Translation
Answer to Question 1: Reflection, Rotation, and Translation (Dilation does NOT apply).
For Question 2, the congruence of the triangles can be justified by any of several theorems, but the relevant theorem depends on how the triangles were matched up.
If we can confirm that corresponding sides are equal and corresponding angles are equal through reflection, rotation, or translation alone, we ideally can justify it using either:
- SSS (Side-Side-Side): if all three sides of \( \triangle ABC \) are equal to \( \triangle EFD \).
- SAS (Side-Angle-Side): if two sides and the included angle of one triangle are equal to the two sides and included angle of the other triangle.
- ASA (Angle-Side-Angle): if two angles and the included side of one triangle are equal to the two angles and included side of the other triangle.
- AAS (Angle-Angle-Side): if two angles and a non-included side of one triangle correspond to two angles and a non-included side of the other triangle.
Given the nature of the transformations applying the congruence relation straightforwardly, the likely correct answer would be:
Answer to Question 2: SAS (if you have two sides and the angle in between them matched through transformations).
However, if angles were just matched without consideration to their sides, ASA or AAS could be possible too, depending on what angles were considered.
Additional considerations:
- HL (Hypotenuse-Leg) only applies to right triangles.
- SSA (Side-Side-Angle) does not justify triangle congruence.
For a clear assertion, you'd typically start with the most direct observation about side and angle relationships that are maintained through the transformations you noted.
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