In this lesson, students draw additional connections between triangle congruence theorems and possible triangle similarity theorems. A key concept is that students need to define what must be true about the sides for “Side-Angle-Side” to stand for something useful for proving that triangles are similar.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Make sure that students are labeling only the information that they know in their diagram, so only a single pair of corresponding angles should be labeled as congruent.

The main idea to draw out of this activity is that knowing that the Side-Angle-Side Triangle Congruence Theorem is true makes it much easier to prove the Side-Angle-Side Triangle Similarity Theorem.

Some students might use the dilation-first argument, and other students might define a specific sequence of rigid motions and a dilation, without mentioning the Side-Angle-Side Triangle Congruence Theorem. Compare the two methods, and discuss how the Side-Angle-Side Triangle Congruence Theorem gives us an opportunity to shorten our proof, by making use of a structure that has already been proven.

In this activity, students continue to understand and prove triangle similarity theorems by adding the Side-Side-Side Triangle Similarity Theorem. 

Some students might use the dilation-first argument outlined in the student response, and other students might define a specific sequence of rigid motions and a dilation without mentioning the Side-Side-Side Triangle Similarity Theorem. This provides an opportunity to compare the two methods and to discuss how the Side-Side-Side Triangle Similarity Theorem gives an opportunity to shorten the proof by making use of structure.