This lesson is optional, as it goes beyond the scope of the standards. While the standards only require students to prove the Angle-Angle Triangle Similarity Theorem, proving other similarity theorems gives students more opportunities to practice creating viable arguments (MP3). In a previous lesson, students saw how the Angle-Side-Angle Triangle Congruence Theorem could be used to prove the Angle-Angle Triangle Similarity Theorem. In this lesson, they see why the Side-Side-Side Triangle Congruence Theorem implies the Side-Side-Side Triangle Similarity Theorem, and why the Side-Angle-Side Triangle Congruence Theorem implies the Side-Angle-Side Triangle Similarity Theorem. Students have a chance to use repeated reasoning (MP8) as they apply what worked in one case to additional cases.
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.
- Prove the Side-Angle-Side and Side-Side-Side Triangle Similarity Theorems (in writing).
- Let’s prove more triangles are similar.
- I can explain why the Side-Angle-Side and Side-Side-Side Triangle Similarity Theorems work.
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 \(A'B'C'\).