Lesson 8

Are They All Similar?

Lesson Narrative

In this lesson, students conjecture and reason about whether all shapes in a certain category must be similar, such as whether all circles, or all rectangles, are similar. Then they prove that all equilateral triangles are similar, and all circles are similar (MP3). The proof that all circles are similar is used again in a subsequent unit, when it is used to rigorously define angle measure, connect angle measure to arc length, and connect degrees to radians.

Students get a chance to apply the theorem they proved in previous lessons, that if two triangles have all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion, then the triangles are similar. This simplifies the proof that all equilateral triangles are congruent. 

Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

Learning Goals

Teacher Facing

  • Critique a proof that uses similarity (in writing).
  • Prove theorems about triangles and other figures using similarity.

Student Facing

  • Let’s prove figures are similar.

Required Materials

Required Preparation

The scientific calculators are for use in the extension problem of the activity “Always? Prove it!”

Learning Targets

Student Facing

  • I can critique proofs that use similarity.
  • I can write proofs using the definition of similarity.

CCSS Standards

Building On


Building Towards

Glossary Entries

  • similar

    One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.

    Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).

    Triangle ABC, rotated and then dilated.