Triangle Congruence

  • Triangle Congruence through Transformations

    8th or HS Geometry


    Can you use transformations to move a triangle onto another if all you know is which measures are the same… and you can't see the triangles?

    Additional Background:

    The Common Core State Standards for Geometry develop the traditional "shortcuts" for proving that two triangles are congruent through transformations, using the definition of congruence that two triangles are congruent if and only if there exist a series of rigid motions mapping one onto the other. This arc uses a series of games, hands-on activities, and interactive online tools to help students:

    • Build up intuition about the relationship between triangle congruence, transformation, and corresponding parts.
    • Experiment with different possible shortcuts and confirm which seem to guarantee congruence every time, and which require additional conditions.
    • Prove the triangle congruence shortcuts (SSS, ASA, and SAS) using transformations/rigid motions.


    This ARC aims to create contexts for students to make connections between transformations, triangle congruence, and the triangle congruence shortcuts. Activities include:

    • dividing irregular shapes into congruent halves
    • constructing triangles with sides the same length
    • using technology and tools to explore shortcuts
    • mapping triangles onto one another based on given information


    ARC authors Max Ray-Riek and Thomas Duarte, and pilot-tester Deidra Baker, reflect on their goals for the ARC, what students learned as Deidra implemented it, and how they might improve the lessons in the future.


    Show students the following image below. Ask them:

    • "Can you add one straight line to the figure that will create two congruent halves?"
    • "How can you convince us that your two halves are congruent?"


    Lesson 1 of 4
    Create two congruent halves from one polygon and use transformations to verify that the two halves are congruent.
    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.

    ARC Assessement

    Technology Resources

    Web Interactives:

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