ABSTRACT Exploring non-Euclidean geometries is a common way for College Geometry instructors to highlight subtleties in Euclidean geometry. The concept of congruence, going undefined or informally defined in multiple axiomatic systems, is particularly susceptible to conflation with the idea of “same measure.” Taxicab geometry provides a context in which congruence (defined as a relationship between figures that can be isometrically mapped to each other) and measure can be disambiguated by investigating the effects of transformations in both Euclidean and Taxicab geometry. This paper describes episodes from our classrooms in which students grappled with their understandings of congruence in Euclidean geometry while completing tasks involving Taxicab geometry.
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