Chapter 38: Similarity: Situation 32 From the MACMTL–CPTM Situations Project

In a geometry class, students were given the diagram in Figure 38.1 depicting two acute triangles, △ABC and △A'B'C', and students were told that △ABC~△A'B'C' with a figure (Figure 38.1) indicating that A'B' = 2ABand m∠B = 75. From this, a student concluded that m∠B' = 150.

By definition, two polygons are similar if and only if their corresponding angles are congruent and their corresponding side lengths are proportional. Thus, similar ?gures may have different sizes, but they have the same shape. The Foci for this Situation incorporate a variety of approaches (geometric, graphical, and symbolic) to shed light on the concept of similarity. The ?rst Focus refutes the claim made in the Prompt by appealing to the de?nition of similar triangles, and the second Focus refutes the claim using an indirect proof that considers the impact of doubling the measures of each of the angles of the original triangle. In Focus 3 and Focus 4, similarity is examined in terms of transformations in general, and dilations in particular. Under a geometric similarity transformation, angle measure is preserved and the ratio of the measures of corresponding distances is constant. Finally, a geometric construction and proof are included that lend further insight into the definition of similarity. In each Focus, the concept of ratio is emphasized, because common ratio lies at the heart of why size, but not shape, may be different for similar figures.