Use the image to answer the question.Two triangles are labeled upper A upper B upper C, and upper F upper E upper D. Describe a sequence of rigid transformations that could be used to show that △ABC≅△DEF. Explain how you know each transformation will result in a congruent figure. Then, explain how this sequence of rigid transformations illustrates the ASA Congruence Theorem.(4 points)

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GPT-4o mini · Answer

To demonstrate that triangles $ \triangle ABC $ and $ \triangle DEF $ are congruent ( $ \triangle ABC \cong \triangle DEF $ ) using a sequence of rigid transformations, a typical approach involves a combination of translations, rotations, and reflections. Below is a suggested sequence of transformations, along with explanations for each step and how they relate to the ASA (Angle-Side-Angle) Congruence Theorem. ### Sequence of Rigid Transformations: 1. **Translation**: - **Description**: Move triangle $ \triangle ABC $ along the plane so that one of its vertices (say vertex A) coincides with the corresponding vertex (vertex D) of $ \triangle DEF $. - **Reason**: A translation shifts the entire triangle without altering its shape or size, maintaining its angles and sides. Thus, triangle $ \triangle ABC $ remains congruent to itself, and now vertex A aligns with vertex D. 2. **Rotation**: - **Description**: Rotate triangle $ \triangle ABC $ about point D (or the new location of A) such that side $ AB $ aligns with side $ DE $. - **Reason**: A rotation also preserves the shape and size of the triangle. After this rotation, angle $ \angle ABC $ will now overlap with angle $ \angle DEF $ since side $ AB $ is aligned with $ DE $. 3. **Reflection** (if necessary): - **Description**: If there is still a difference in orientation between $ \triangle ABC $ and $ \triangle DEF $, you may reflect triangle $ \triangle ABC $ over an appropriate line (like the line containing side $ DE $). - **Reason**: A reflection will flip the triangle over the line, ensuring that the correct orientation is achieved. This preserves the lengths of sides and measurements of angles, thus maintaining congruence. ### Explanation of Congruence: - Each of these transformations—translation, rotation, and reflection—is a rigid transformation, meaning they do not change the size or shape of the figure. - Rigid transformations preserve the angles and lengths of sides, which is crucial for demonstrating congruence. ### ASA Congruence Theorem: The ASA Congruence Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. - **Application**: By using translations and rotations (and potentially reflections), we align the two triangles in such a way that: 1. Two angles of $ \triangle ABC $ are congruent to the two angles of $ \triangle DEF $ (for instance, if $ \angle A $ ≅ $ \angle D $ and $ \angle B $ ≅ $ \angle E $). 2. The side between those two angles (side $ AB $ ≅ side $ DE $) remains congruent through the transformations. By confirming the relationships established through the transformations, we establish that $ \triangle ABC \cong \triangle DEF $ via the ASA Congruence Theorem, demonstrating the congruent nature of the two triangles geometrically.