To show how AAS (Angle-Angle-Side) follows from the definition of congruence, we typically need to demonstrate that two triangles can be made to coincide through rigid transformations. Rigid transformations include translations, rotations, and reflections.
Since the specifics of triangles D E F and A double prime B double prime C double prime (A", B", C") are not provided, I will describe a general approach to answering the question about mapping △A"B"C" to △DEF.
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Identify Corresponding Angles and Sides: First, determine which angles and sides in triangle A" B" C" correspond to which angles and sides in triangle DEF. According to the AAS theorem, we typically have two angles and the included side to show congruence.
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Determine the Rigid Transformation:
- Translation: If the triangles are in different positions, you might start by translating triangle A" B" C" so that one of its vertices aligns with the corresponding vertex in triangle DEF.
- Rotation: If the triangles need to be oriented similarly, apply a rotation around the corresponding vertex.
- Reflection: If the triangles are mirror images of each other or need to be flipped, you may apply a reflection.
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Perform the Transformation: Carry out the transformation you have identified and verify that after applying these transformations, triangle A" B" C" coincides with triangle DEF, showing that they are congruent.
For a specific transformation, you would analyze the given triangles, determine the necessary translations, rotations, or reflections needed to align them, and describe that process.
Example conclusion: "A rotation of 180 degrees around point A" followed by a translation to align vertex A" with vertex D would map triangle A" B" C" onto triangle DEF, demonstrating the congruence through AAS."
Please reference the actual image to determine the exact nature of these transformations based on the orientations and positions of the triangles.