Correspondence and Transformations
The figure to the right represents a rotation of △ 𝐴𝐵𝐶 80° around vertex 𝐶. Name the triangle formed by the image of △ 𝐴𝐵𝐶. Write the rotation in function notation, and name all corresponding angles and sides.
In the Opening Exercise, we explicitly showed a single rigid motion, which mapped every side and every angle of △ 𝐴𝐵𝐶 onto △ 𝐸𝐹𝐶. Each corresponding pair of sides and each corresponding pair of angles was congruent. When each side and each angle on the pre-image maps onto its corresponding side or angle on the image, the two triangles are congruent. Conversely, if two triangles are congruent, then each side and angle on the pre-image is congruent to its corresponding side or angle on the image.
𝐴𝐵𝐶𝐷 is a square, and 𝐴𝐶 is one diagonal of the square. △ 𝐴𝐵𝐶 is a reflection of △ 𝐴𝐷𝐶 across segment 𝐴𝐶. Complete the table below, identifying the missing corresponding angles and sides.
a. Are the corresponding sides and angles congruent? Justify your response.
b. Is △ 𝐴𝐵𝐶 ≅ △ 𝐴𝐷𝐶? Justify your response.
Each exercise below shows a sequence of rigid motions that map a pre-image onto a final image. Identify each rigid motion in the sequence, writing the composition using function notation. Trace the congruence of each set of corresponding sides and angles through all steps in the sequence, proving that the pre-image is congruent to the final image by showing that every side and every angle in the pre-image maps onto its corresponding side and angle in the image. Finally, make a statement about the congruence of the pre-image and final image.
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