To determine the two transformations needed to show that the triangles are congruent, we would typically analyze the specific details of the triangles in question. However, since the specific triangles are not provided here, I can provide a general approach:
- For congruence through transformations: The transformations that often prove triangles are congruent include:
- Translation: Moving the triangles without rotation or resizing.
- Rotation: Turning the triangles around a specific point.
- Reflection: Flipping the triangles over a line.
The most common pairs would be:
- Rotation and Reflection.
- Translation and Reflection.
If we were to choose them without any visual aids or specifics, a common pair might be Translation and Reflection or Rotation and Reflection, depending on their orientations.
For the second question concerning which congruency theorem is justified by the transformations, the possible choices are:
- SSS (Side-Side-Side): All sides of one triangle are equal to all sides of another triangle.
- ASA (Angle-Side-Angle): Two angles and the included side are equal.
- AAA (Angle-Angle-Angle): All angles are equal (does not prove congruence).
- HL (Hypotenuse-Leg): Used for right triangles based on the hypotenuse and one leg.
- AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
The correct answer for the congruency theorem would depend on the information given about the triangles and which sides or angles are established to be congruent through the transformations. If rotations or reflections lead us to side-sides or angle relations that maintain congruence, SSS or ASA might be appropriate if the included side or all sides are established.
Without the context of the specific transformations you performed, the most general answers for transformations and the congruency theorem would likely be Translation and Reflection with SSS or ASA, depending on the established relationships post-transformation.
If more specific details are available, please provide them for a more accurate response!