Triangle Congruence Lesson 1

  • Congruent Halves

    Lesson 1 of 4
    8th or HS Geometry

    40-60 minutes


    Create two congruent halves from one polygon and use transformations to verify that the two halves are congruent.



    Begin class by having students discuss what it means for two figures to be congruent.

    • Students might have a wide range of answers, from "exactly the same" to "same size/same shape" to "you can put them on top of each other with no overlap."
    • If students aren't confident in what "congruent" means, show two figures that are congruent and two that are not and ask students to explain why the two figures in Image 1 are congruent and the two figures in Image 2 are not congruent. (SMP 6)
    • Then ask students, "How can you show two figures are congruent?"
      • Student answers might range from "measure them" to "see if they're exactly the same" to "put one on top of the other and make sure they overlap exactly."
    • If students don't bring up the idea of "putting them on top of each other," tell students, "Today we are going to be exploring whether some complicated shapes are congruent. Why do you think you were given patty paper?"
      • Students may or may not connect the patty paper to the idea of rigid motions to transform one shape onto the other, but they will likely guess that the patty paper can be used to "put one shape on top of the other".
    • Tell students, "When working with these shapes and the patty paper, I'd like us to go beyond just putting the tracing on top of the shape and showing that they match up exactly. You'll be describing the translations, rotations, and reflections that you used to move one shape onto the other. In case you need it, here is a reminder of what those words mean." (SMP 6))
    • For English language learners, it may be useful to print and have on a "Word Wall" the images you show of a rotation, reflection, and translation.


    This activity has two distinct parts.

    • For Part 1 students are trying to divide figures into two congruent figures by adding segments to the figure.
    • For Part 2 students are using rigid transformations to explain how they know they successfully cut the figure into two congruent figures.

    Part 1

    • Give students the Lesson 1 Student Activity Sheet "Where Are The Congruent Halves?" (download this from the Materials section above).
    • During Part 1 students may explore a guess and check strategy when trying to divide the two figures. Try to find one or more students who uses a different strategy to divide the figure and invite them to share their process with the class. An example may be trying to find a corresponding part that has to be congruent and working from there.
      • "Explain how you determined where to divide the figure." (SMP 1, SMP 3)
      • "What do you look for when trying to divide the figure in congruent parts?" (SMP 1, SMP 3)
      • "Is it possible to have more than one answer? Can you show me an example?" (SMP 1, SMP 3)
    • It should be emphasized to students that they can use their patty paper to check if their solutions are correct (SMP 5). Have students share results for the more difficult problems (e.g., 5, 6, and 7). Invite other students to show what rigid transformations they could use to show that their classmates' halves are indeed congruent.
      • "Do you think those figures are congruent? Can you show the series of transformations that gets them to line up exactly?"
      • Prompt students to use the transformation vocabulary: "When you say 'moving', 'flipping' or 'turning', what do we call that in mathematics?" (SMP 6)
      • Prompt high-school students to define the transformations precisely:
      • For translations, students should define the transformation in terms of a point in the pre-image and image, e.g. I translated A to A'.
      • To prompt students for clarity, ask "Can you tell me where Point A will go? Then you translated from Point A to ____"
      • For rotations, students should define the transformation in terms of a center of rotation and an angle of rotation.
      • To prompt students for clarity, ask "What point did you rotate around? How did you know to stop rotating?" Help students define the angle of rotation in terms of points in the figure, it it's not obviously 90º, 180º, etc.
      • For reflections, students should define the transformation in terms of a line of reflection.
      • To prompt students for clarity ask, "What line did you reflect over? What line is the 'mirror'?"

    Part 2

    • During Part 2, students will record the process they used to verify the two figures are congruent. The precision of the language used to explain this process should change depending on the grade level this activity is used in. If this task is used in eighth grade, then students can simply state, "I used a translation, then a rotation." If this task is used in high school, then students should be encouraged to label points on the diagram and precisely describe the process they used to verify congruence.
      Possible extension questions include:
      • "Is it possible to show the figures are congruent using fewer transformations?"
      • "Is it possible to show the figures are congruent using a rotation?"


    Think - Write - Pair - Share

    Michael said the he wasn't able to find a sequence of rigid motions from Triangle ABC to Triangle A'B'C'. What do you think that means? Explain your reasoning.

    Teacher Reflection

    Elicit and use evidence of student thinking:

    • What informal or formal definitions of congruence did your students have at the beginning of the lesson?
    • Informally, we often think of congruence as meaning either "same size and same shape" or "match up exactly". How might each informal definition connect to the formal definition of congruence as "two shapes are congruent if there exist a series of rigid motions mapping one to the other?"
    • Based on what your students struggled with and had success with, do you think that they -
      • have a working definition for "congruent" that includes some idea of parts matching up and some idea of being able to superimpose the two images and see no "gaps"?
      • can look for corresponding parts of figures to help them estimate whether two figures are likely to be congruent?
      • can, eventually, find transformations that will map any two congruent figures onto one another?
      • are getting better at describing those transformations precisely? If so, they are ready to keep moving forward to the next lesson!

      >Support productive struggle in learning mathematics:

      • What parts, if any, of Lesson 1 did your students struggle with?
        • Finding likely ways to divide the figure to make congruent halves?
        • Finding a series of transformations that map one half onto the other?
        • Precisely describing the series of transformations they found?
      • How did you support students so that they had to grapple with rigorous mathematics, but had what they needed to persevere?
      • How are you celebrating student struggle and perseverance?

      Related Material

      Mathematics Teaching in the Middle School, Samuel Obara

      Other Lessons in This Activity

      Lesson 2 of 4
      Students grapple with congruence through rigid transformations, then conjecture a "shortcut" set of conditions that also ensure congruence..
      Lesson 3 of 4
      What is the minimum info needed to prove two triangles are congruent? Play a game where the less info you give, the more points you get!
      Lesson 4 of 4
      Through a game, students build up their skills at transforming general triangles with specific starting conditions.
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  • Essential Question(s)

    • How can you be confident that two figures are congruent? How can you communicate the process you used to determine that two figures are congruent?


    CCSS, Content Standards to specific grade/standard

    • 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.

    CCSS, Standards for Mathematical Practices

    • SMP 1 Make sense of problems and persevere in solving them.
    • SMP 3 Construct viable arguments and critique the reasoning of others.
    • SMP 5 Use appropriate tools strategically.
    • SMP 6 Attend to precision.

    PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

    • Support productive struggle in learning mathematics.
    • Elicit and use evidence of student thinking.