Similarity

New York State Common Core Math Geometry, Module 2, Lesson 14

Worksheets for Geometry

Student Outcomes

• Students understand that similarity is reflexive, symmetric, and transitive.
• Students recognize that if two triangles are similar, there is a correspondence such that corresponding pairs of angles have the same measure and corresponding sides are proportional. Conversely, they know that if there is a correspondence satisfying these conditions, then there is a similarity transformation taking one triangle to the other respecting the correspondence.

Similarity

Classwork

Example 1

We said that for a figure 𝐴 in the plane, it must be true that 𝐴~𝐴. Describe why this must be true.

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Example 2

We said that for figures 𝐴 and 𝐵 in the plane so that 𝐴~𝐵, then it must be true that 𝐵~𝐴. Describe why this must be true.

Example 3

Based on the definition of similar, how would you show that any two circles are similar?

Example 4

Suppose △ 𝐴𝐵𝐶 ↔ △ 𝐷𝐸𝐹 and that, under this correspondence, corresponding angles are equal and corresponding sides are proportional. Does this guarantee that △ 𝐴𝐵𝐶 and △ 𝐷𝐸𝐹 are similar?

Example 5

a. In the diagram below, △ 𝐴𝐵𝐶~ △ 𝐴′𝐵′𝐶′. Describe a similarity transformation that maps △ 𝐴𝐵𝐶 to △ 𝐴′𝐵′𝐶′.
b. Joel says the sequence must require a dilation and three rigid motions, but Sharon is sure there is a similarity transformation composed of just a dilation and two rigid motions. Who is right?

Lesson Summary

• Similarity is reflexive because a figure is similar to itself.
• Similarity is symmetric because once a similarity transformation is determined to take a figure to another, there are inverse transformations that can take the figure back to the original.

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