Lesson 10

The purpose of this Math Talk is to elicit strategies and understandings students have for triangle congruence shortcuts. These understandings help students recall what they learned in a previous unit and will be helpful later in this lesson when students will need to be able to use these shortcuts in proofs of similarity. In this activity, students have an opportunity to notice and make use of structure (MP7) as they choose which criteria apply to a certain case. Since the images below have multiple ways they could be proven to be congruent, comparing ideas brings out structure.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have 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?”

In a previous lesson, students studied the connection between the Angle-Side-Angle Triangle Congruence Theorem and the Angle-Angle Triangle Similarity Theorem. This activity invites students to conjecture about the Side-Angle-Side Triangle Similarity Theorem. 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 only labeling the information 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 structure.

In a previous lesson, students studied the connection between the Angle-Side-Angle Triangle Congruence Theorem and the Angle-Angle Triangle Similarity Theorem. In a previous activity, students proved the Side-Angle-Side Triangle Similarity Theorem. This activity gives students an opportunity to apply what they learned about that proof to prove 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 discuss how the Side-Side-Side Triangle Similarity Theorem gives an opportunity to shorten the proof by making use of structure.