![]() Note that if $A = D$ for the original pair of triangles then this first step is not necessary. The picture below shows the image of $\triangle ABC$ after this translation, denoted $\triangle A^\prime B^\prime C^\prime$: We can choose a translation that maps $A$ to $D$ while preserving all side lengths and angles of $\triangle ABC$. We work with the given example of two triangles $ABC$ and $DEF$ with congruent corresponding sides and angles but the method presented here will work for any pair of triangles with congruent corresponding angles and sides: we have made indications throughout how the given construction would apply in general. In other words, corresponding parts of congruent triangles are congruent. In addition, they will need to match their visual ideas for how to show the triangle congruence in part (a) with the mathematical definitions of the different transformations (translations, rotations, and reflections) in part (b). Students will need to keep track of what is given and what needs to be shown since these two trade places in parts (a) and (b). The main mathematical practice closely associated with work on this task is MP2, Reason Abstractly and Quantitatively. In addition, working with transformations helps to build visual intuition and to identify important structure, namely symmetry, in nature and art. The properties of transformations, taken for granted in this approach to geometry, serve to make these unstated hypotheses explicit. These congruence criteria are at the heart of Euclidean geometry but providing good proofs either requires a lot of work building up from basic axioms (as is done in college modern geometry courses) or requires use of visual intuition and unstated hypotheses (as was done in Euclid's time). One of the advantages to working with the rigid motion definition of congruence is that it allows relatively quick and clean proofs of the basic triangle congruence criteria (see tasks illustrating G-CO.8 for more details). In particular, there is a sequence of rigid motions mapping one triangle to another if and only if these two triangles have congruent corresponding sides and angles. ![]()
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