In this activity students prove that all segments of the same length are congruent. First, students disprove a conjecture that all segments are congruent. Then, students work on their proof of the modified conjecture. This proof leads directly to the triangle congruence proofs in subsequent lessons.

Students may forget to specify that the points will coincide since the segments are the same length. Remind them of the counterexample in the Launch.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their proofs. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

Remind students they can use the words and structure from the display of sentence frames for proofs. If time allows, display these prompts for feedback: 

• “What is the directed line segment of the translation?”
• “What is the center and angle of rotation?”
• “How do you know that these points will coincide?” 

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.

Invite students to share pieces of the proof until the whole class agrees that the proof is sufficiently detailed and convincing. Help students determine when they should refer to rays versus segments to solidify the idea that the segments aren’t congruent until they have used the fact that they are the same length.

• Translate segment by the directed line segment from to . 
• Points and coincide because we defined our transformation that way.
• Rotate segment using as the center so that ray coincides with ray . 
• Translation and rotation both preserve distance, so segment is the same length as segment , which means segment is the same length as segment . 
• Since points and are the same distance along the same ray from , they have to be in the same place. 
• Therefore, there is a rigid motion that takes to , so the segments must be congruent.
   

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

If two segments have the same length, then they are congruent. (Theorem)

In this partner activity, students take turns using their new theorem about congruent segments to prove figures made of congruent segments are congruent. As students trade roles listening and explaining their thinking, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Invite students to write sentence frames for the new transformation they used in their proofs. Add them to the display of sentence frames for proofs. This display should be posted in the classroom for the remaining lessons within this unit. More will be added to the display throughout the unit. An example template is provided with the blackline master for this lesson.

Transformations:

• Segments  and  are the same length, so they are congruent. Therefore, there is a rigid motion that takes  to . Apply that rigid motion to .