# Lesson 9

Conditions for Triangle Similarity

### Lesson Narrative

In previous lessons in this unit, students were encouraged to consider proving triangles similar by first dilating one of the triangles by a strategically chosen scale factor, and then using one of the triangle congruence theorems to prove that the resulting triangle was congruent to the given triangle, and thus the original triangle was similar to the given triangle.

In this lesson, that work pays off as students use the Angle-Side-Angle Triangle Congruence Theorem as the basis for proving the Angle-Angle Triangle Similarity Theorem. The warm-up invites students to notice structure (MP7) and strategically make use of the Angle-Side-Angle Triangle Congruence Theorem as they prove that two triangles are similar. Then students have the opportunity to build intuition about the Angle-Angle Triangle Similarity Theorem before proving it. The lesson culminates with students exploring how many angles are needed, and which angles are needed, to be sure we can use the Angle-Angle Triangle Similarity Theorem to prove two triangles are similar.

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

• Prove the Angle-Angle Triangle Similarity Theorem (in writing).

### Student Facing

• Let’s prove some triangles similar.

### Student Facing

• I can explain why the Angle-Angle Triangle Similarity Theorem works.

Building On

Addressing

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 $$A'B'C'$$.