I need to prove that if triangles 1 and 2 are congruent and both lying in $\mathbb R^2$, there is an isometry mapping triangle 1 to triangle 2, where the definition of an isometry is here taken to be $Ox+a$ where $O$ is an orthogonal matrix and $a$ is just some 2-vector. This is intuitively obvious, but how might I prove it rigorously?
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You should try to do a specific example yourself. Draw two congruent triangles and try to find an isometry. Then use this to construct a general proof.