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?
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:
This ARC aims to create contexts for students to make connections between transformations, triangle congruence, and the triangle congruence shortcuts. Activities include:
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:
How do the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions?
In more student-friendly language:
“If we've been proving that triangles are congruent by showing how we can rotate, reflect, and translate one onto the other, why is it now sufficient to just show that all the side lengths match up, or just show that two angle pairs and one pair of sides match up?”
CCSS, Content Standards to Domain Level
CCSS, Standards for Mathematical Practices
New Editors: Deidra Baker, Thomas Duarte, Max Ray-Riek
This was an original lesson plan -- the Illuminations Interactive and MTMS articles that inspired it are cited as resources in the relevant lessons.