This Math Talk focuses on using the idea of congruence to show that two triangles are similar. It encourages students to think about the definitions of congruence and similarity and to rely on what they know about triangle congruence to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students examine triangle similarity theorems.

To show congruence between triangles, students need to look for and make use of structure (MP7).

The goal of this discussion is to review students’ strategies for finding similarity and congruence between triangles.

To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone use the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

If no students bring up the Angle-Side-Angle Triangle Congruence Theorem, ask students how they might use the triangle congruence theorems that they know. Then ask students, “If a dilation takes triangle to triangle , and triangle is congruent to triangle , how do we know that triangle is similar to triangle ?” (After the dilation, the figures are congruent which means there is a sequence of rigid motions that takes onto triangle . The same sequence of rigid motions following the dilation to make triangle shows that triangle and triangle are similar by the definition of similarity.)

While the Angle-Angle Triangle Similarity Theorem follows logically from the Angle-Side-Angle Triangle Congruence Theorem (dilate one triangle by a scale factor determined by the ratio of the sides between the known corresponding angle pairs, and the dilated triangle will be congruent to the target by the Angle-Side-Angle Triangle Congruence Theorem, so the original triangle must be similar to the target), it can still be unintuitive.

Students experienced the Angle-Angle Triangle Similarity Theorem briefly in a previous unit when they looked at examples of triangles constructed so that their angles were the same but their side lengths were undetermined. In middle school, students confirmed the Angle-Angle Triangle Similarity Theorem informally. However, their proofs until this point have tended to rely on knowing something about the proportionality of side lengths.

Students might be unsure that the same scale factor will work for all three pairs of corresponding sides when all they know is the measure of two angles of the triangle. It is a surprising result when you put it that way. This activity starts by building students’ intuition about the Angle-Angle Triangle Similarity Theorem, to motivate the proof of the theorem.

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

This activity expands upon the Angle-Angle Triangle Similarity Theorem by requiring the use of the Triangle Angle Sum Theorem to find two pairs of congruent angles. Once similarity is established, students use the scale factor to determine missing side lengths on the triangles.

This is the first time Math Language Routine 5: Co-Craft Questions is suggested in this course. In this routine, students are given a context or situation, often in the form of a problem stem (for example, a story, image, video, or graph) with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?" The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.