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 point-by-point 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.
- Justify (orally and in writing) that two triangles are congruent if and only if all corresponding sides and angles are congruent.
- Let’s use transformations to be sure that two triangles are congruent.
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.
- I can explain why if all the corresponding sides and angles of two triangles are congruent, then the triangles are congruent.
For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.
In the figure, segment \(AB\) corresponds to segment \(DE\), and angle \(BCA\) corresponds to angle \(EFD\).