Discovering Triangle Congruence Shortcuts
Lesson 4 of 10
Objective: Students will be able to conjecture about the sets of three sides and angles that will guarantee triangle congruence.
I use this warm-up because it forces students to apply their understanding of the angles of isosceles triangles in an unconventional way. This problem can yield a really rich whole-class discussion since there are several ways to think and reason through the problem. What I like to do for the first problem is write out three different possible equations that represent the geometric relationships in the problem; I then ask students to take a moment to consider what ideas the equations represent and to justify how they know—this really gives students an opportunity to look for structure in the work (MP7) and to defend their thinking using precise vocabulary (MP3).
I launch this Investigation by leading a short whole-class discussion around triangle congruence. I offer a simple definition for congruence—all corresponding sides and angles are congruent—an idea that makes sense to students but sounds like a rather time consuming process. This is when I plant the idea of looking for shortcuts, which motivates the investigation. The Investigation is a task to see which of the six permutations of three sides and three angles will guarantee triangle congruence.
I ask students to work in pairs and to use their constructions skills (MP5) to make sense of the investigation. They are given a set of sides and angles from which they are to determine the number of triangles possible to construct—if they can construct one and only one triangle, the pair should conjecture that the shortcut might guarantee triangle congruence; if they can construct more than one triangle, the pair should conjecture that the shortcut does not guarantee triangle congruence. In this investigation, pairs will deal with SSS, SAS, ASA, SSA; they will come across AA later in the lesson, and they will prove AAS in the next lesson. Throughout this investigation, students are trying to look for and express regularity in repeated reasoning, considering which combination of three sides and angles can guarantee triangle congruence (MP8).
During the debrief of this activity, I choose four different students’ constructions for SSS, SAS, ASA, and SSA to share using the document camera. The rest of the class checks their work against the constructions shown with the goal of showing a counterexample, which would disprove the conjectures students had written during the investigation. After we go through this process, we formally debrief on our note takers, which is a routine for our class.
In this Exit Ticket, I display four pairs of triangles from the station work. Students choose any one of the pairs of triangles to prove congruent. This exit ticket gives me a way to formatively assess my students' understanding about triangle congruence criteria and whether they can prove triangles are congruent.