This lesson introduces students to proofs of triangle congruence using transformations. Prior to this lesson, students have focused on finding the transformation or sequence of transformations that appear to take a given figure onto another. Their practice with pointbypoint transformation will be particularly relevant. In this lesson, they grapple with the idea that the right set of transformations will work for any set of triangles with the right congruent corresponding parts, regardless of position and orientation.
Writing proofs using transformations requires constructing arguments for why the sequence of moves is guaranteed to line up the vertices and sides exactly (MP3). As students progress through the unit, they will have more opportunities to create their own proofs as well as see models of increasingly formal language with which to express their reasoning. Creating a bank of statements and reasons for students to draw on can reduce the cognitive demand of writing proofs with formal language, so students can focus on the logical coherence. A template is provided with the blackline masters for this lesson.
Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview

3.1 Warmup: True or . . . Sometimes True?: Triangles (5 minutes)
 Digital applet in this activity

3.2 Activity: Invisible Triangles (20 minutes)
 Includes "Are you Ready for More?" extension problem
 3.3 Activity: Why Do They Coincide? (10 minutes)
 Lesson Synthesis
 3.4 Cooldown: Reflecting on Proof (5 minutes)
Learning goals:
 Justify (orally and in writing) that two triangles are congruent if and only if all corresponding sides and angles are congruent.
Learning goals (student facing):
 Let’s use transformations to be sure that two triangles are congruent.
Learning targets (student facing):
 I can explain why if all the corresponding sides and angles of two triangles are congruent, then the triangles are congruent.
Required materials:
 Geometry toolkits
 Preprinted cards, cut from copies of the blackline master
Required preparation:
 For the Invisible Triangles activity: Separate the transformer cards from the triangle cards and give each group one transformer card and one set of three triangle cards.
 If feasible, provide each group of 2 with a folder or other divider so students can’t see each other’s desktops.
 Create a display of sentence frames for proofs.
 This display should be posted in the classroom for the remaining lessons within this unit.
 There is a blackline master with the final version; it will be built over several lessons.
Standards:
 This lesson builds on the standards: CCSS.8.G.A.1.aMS.8.G.1aCCSS.8.G.A.1.bMS.8.G.1bCCSS.HSGCO.A.5MS.GCO.5CCSS.HSGCO.B.6MS.GCO.6MO.8.GM.A.1aMO.8.GM.A.1bMO.G.CO.A.5MO.G.CO.B.6
 This lesson builds towards the standards: CCSS.HSGCO.B.8MS.GCO.8MO.G.CO.B.7
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.