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 that all circles are similar, and they critique an argument with faulty reasoning that all rectangles 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 that 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.