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.

Storyboard

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

Audio

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.

Hook

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?"