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?"

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?”

8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

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.

HSG.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

HSG.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

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.

SMP 7 Look for and make use of structure.

Contributors

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.