Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
This activity invites students to think carefully about the definition of congruence. Students can use any rigid motion to demonstrate all points are congruent.
Math Community
After the Warm-up, display the Math Community Chart. Remind students that norms are agreements that everyone in the class shares responsibility for, so it is important that everyone understands the intent of each norm and can agree with it. Tell students that today’s Cool-down includes a question asking for feedback on the drafted norms. This feedback will help identify which norms the class currently agrees with and which norms need revising or removing.
Launch
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Activity
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If is a point on the plane and is a point on the plane, then is congruent to .
Try to prove this claim by explaining why you can be certain the claim must be true, or try to disprove this claim by explaining why the claim cannot be true. If you can find a counterexample in which the “if” part (hypothesis) is true, but the “then” part (conclusion) is false, you have disproved the claim.
Activity Synthesis
Invite students to share. If students only suggest translation, ask if it is possible to use another transformation. If rotation is not mentioned by students, there is no need to continue to push for that solution.
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.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing and speaking.
Launch
Display the conjecture, and invite students to prove or disprove it.
“If is a segment in the plane and is a segment in the plane, then is congruent to .”
Give students 1 minute of quiet work time. Invite them to refer to the instructions in the Warm-up for how to prove or disprove an if-then statement.
Select a student who has drawn a counterexample to share. Invite a student to rephrase the conjecture so it will always be true. (If is a segment in the plane and is a segment in the plane with the same length as , then is congruent to .)
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).
Launch
Arrange students in groups of 2.
Representation: Access for Perception. Read the statements about the two zigzags aloud. Students who both listen to and read the information will benefit from extra processing time. Supports accessibility for: Language, Attention
Lesson Synthesis
Display two congruent zigzags with several segments each. Invite students to contribute one statement at a time to prove the zigzags are congruent. Refer them to the sentence frames when they get stuck.
Student Lesson Summary
If two figures are congruent, then there is a sequence of rigid motions that takes one figure onto the other. We can use this fact to prove that any point is congruent to another point. We can also prove segments of the same length are congruent. Finally, we can put together arguments to prove entire figures are congruent.
These statements prove is congruent to .
Segments and are the same length, so they are congruent. Therefore, there is a rigid motion that takes to . Apply that rigid motion to figure .
If necessary, reflect the image of figure across to be sure the image of , which we will call , is on the same side of as .
must be on ray since both and are on the same side of and make the same angle with it at .
Since points and are the same distance along the same ray from , they have to be in the same place.
Therefore, figure is congruent to figure .
Student Response
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Building on Student Thinking
Activity
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Prove the conjecture: If is a segment in the plane and is a segment in the plane with the same length as , then is congruent to .
Student Response
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Building on Student Thinking
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.
Activity Synthesis
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)
, so .
Activity
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Here are some statements about two zigzags. Put them in order to write a proof about figures and .
1: Therefore, figure is congruent to figure .
2: must be on ray since both and are on the same side of and make the same angle with it at .
3: Segments and are the same length, so they are congruent. Therefore, there is a rigid motion that takes to . Apply that rigid motion to figure .
4: Since points and are the same distance along the same ray from , they have to be in the same place.
5: If necessary, reflect the image of figure across to be sure the image of , which we will call , is on the same side of as .
Take turns with your partner stating steps in the proof that figure is congruent to figure .
Figures A B C D and E F G H, with different orientations. Segments A B and E F each have 1 tick mark, B C and F G each have 2 tick marks, C D and G H each have three tick marks. Angles A B C and E F G each have 1 tick mark, Angles B C D and F G H each have two tick marks.
Student Response
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Building on Student Thinking
Activity Synthesis
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 .
Standards Alignment
Building On
Addressing
HSG-CO.B.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
