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.
- Critique a proof that uses similarity (in writing).
- Prove theorems about triangles and other figures using similarity.
- Let’s prove figures are similar.
The scientific calculators are for use in the extension problem of the activity “Always? Prove it!”
- I can critique proofs that use similarity.
- I can write proofs using the definition of similarity.
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 \(A'B'C'\).