In a previous lesson, students justified that, if all pairs of corresponding sides and angles are congruent, then the two triangles must be congruent. This lesson invites the question, “How much information is needed to guarantee that two triangles are congruent?” Students start to answer this question in an Info Gap. When students request information about sides and angles, students must attend to precision (MP6). In subsequent activities, students study character dialogues from the Info Gap to highlight the ambiguity of requesting two sides and an angle or three angles.
Throughout this lesson, students are developing and testing conjectures about how much information they need to prove triangles are congruent (MP8). They will then write these proofs over the next several lessons. The process of experiment, conjecture, test, and adjust or prove is the essence of doing mathematics.
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.
- Generate (orally) conjectures about sufficient conditions to prove that triangles are congruent.
- Let’s figure out if there are shortcuts for being sure two triangles are congruent.
Create a display labeled “Triangle Congruence Criteria Conjectures” for the lesson synthesis.
- I can write conjectures about what I need to know to prove two 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 \(EFD\).