Lesson 2

The purpose of this Math Talk is to elicit strategies and understandings students have for connecting corresponding parts to congruence. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to name corresponding parts accurately and use them in proofs. While participating in this activity, students need to be precise in their word choice and use of language (MP6).

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.

If students struggle, encourage them to mentally “connect the dots,” or use their finger to trace around the figure in the order indicated by each set of letters. Students can also use a highlighter to highlight the first pair of corresponding sides (according to how the figures are named) the same color, then highlight the second pair with a second color, etc.

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

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same reasoning but would explain it differently?”
• “Did anyone think of it in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

Emphasize that yes, all the figures in the pictures are congruent, but we are asking about the figures named by the points. Invite a student to highlight for all to see the corresponding parts of triangles \(JKL\) and \(QRS\) as named (use colored pencils to draw over segments \(JK\) and \(QR\) in one color, segments \(KL\) and \(RS\) in a second color, and segments \(JL\) and \(QS\) in a third color). Ask students if the corresponding parts as named are congruent.

In a previous lesson, students justified that two figures being congruent guarantees that all pairs of corresponding parts are congruent. In this activity, students explore a different direction. If any pair of corresponding parts is not congruent, then the two figures cannot be congruent.

Monitor for students who:

• focus on corresponding parts having the same measures
• try lining up the triangles (either physically, with tracing paper, or mentally)
• try to use the point-by-point method to prove it’s possible to use rigid transformations to take the figures onto one another

This activity previews the triangle congruence criteria. Here, they are given three pairs of congruent corresponding parts (two side lengths and one angle measurement), which is not enough information to be sure that all triangles with these measurements are congruent (this is an example of the Side-Side-Angle case).

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

Students may believe all 3 triangles are congruent. If this happens, invite them to use available tools (tracing paper, compass, or ruler) to check.

The key point of this discussion is that triangle \(ACE\) cannot be congruent to triangle \(PQR\) because at least one of the corresponding parts is not congruent.

Select multiple students to explain their reasoning.

• students who answered either question in terms of corresponding parts
• students who say the triangles do or don’t line up
• students who used the point-by-point method to prove it’s not possible to use rigid transformations to take \(PQR\) onto \(ACE\)

If no student used the point-by-point method, invite students to try to write a sequence of rigid motions to take \(PQR\) onto \(ACE\). Then, ask them which step they get stuck on. This builds toward the proofs they will write for triangle congruence theorems in subsequent lessons.

The mathematical goal of this activity is to name corresponding parts, and recognize which parts correspond based on the congruence statement. The first figure is designed so students can see that the named angles do not correspond. The second figure is visually challenging, but the answer can be determined by looking at how the figures are named. The third figure has no diagram.

Monitor for students who see how to use the naming of figures to find the answer without a diagram.

Suggest that students who struggle more than is productive redraw each figure in the same orientation. Direct them to the order of the letters in the congruence statement to support them with labeling.

If most students struggled or used a diagram, invite a student to share how they used the naming of the figure to find the answer without a diagram. Tell students there is no one best strategy, but that this is an option.