Proofs with Triangle Congruence Shortcuts
Lesson 5 of 10
Objective: Students will be able to apply the triangle congruence shortcuts (SSS, SAS, ASA, AAS) to write proofs about special triangles and parallelograms that involve triangle congruence.
Warm-Up: SSS and SAS
I use this warm-up to assess students’ understanding of what they would need to write up in a proof to prove or disprove triangles congruent. In this warm-up, I intentionally show two pairs of triangles that do not appear to be congruent—I do this because I need students to avoid making quick assumptions to instead build good habits around identifying what is given to draw logical conclusions.
As I have done with other proofs in the last few units, I try to find student volunteers who are willing to take risks and put their proofs out there for critique. At this point in the unit, it is really good to find students who are willing to share their work and receive feedback on their proofs publicly—this gives the entire class a more concrete understanding of the characteristics of good proofs (using the necessary given information, ending with a logical conclusion, having sufficient evidence along the way, etc.) and an idea of how they can self-assess their proof writing (MP1, MP3).
After the Warm-Up we transition into a sequence of:
- individual time
- group discussion
- whole-class discussion
To begin, I give students a few moments to review the “notes” we took at the end of the last lesson, where we concluded that SSS, SAS, ASA guarantee triangle congruence and SSA did not.
Next, I ask students to consider whether AAA can guarantee triangle congruence. In order for this discussion to be effective, I use the convince yourself, convince a friend, convince a skeptic instructional routine (MP1, MP3). In this discussion framework, students have individual time to formulate their own point of view, time to “convince a friend” (another person in their group), and then time to “convince a skeptic” where trying to formulate a convincing argument to persuade others to believe them. Later, when individual students present their ideas about AAA, I ask the entire class to take on the role of the skeptic, which makes it safe for anyone in the audience to ask questions, ask for clarity, ask for re-explanation, or ask for additional examples (MP6).
After the AAA whole-class discussion, I then ask students to consider whether they think AAS could guarantee triangle congruence. Some students might share out an initial “hunch” (for example, that the pair of congruent sides would guarantee both triangles have to be not only similar, but also congruent) but as a class, we know that we need to move beyond hunches to actually prove something about AAS. I lead the whole class through this proof, which we then record in our Triangle Congruence note taker.
Since there has been lots of group and whole-class discussion, I give students the opportunity to self-assess their understanding of the triangle congruence shortcuts. I pass out four pairs of triangles and ask students to determine whether the triangles are congruent and to justify how they know. After students have had time to work, I review the answers with the whole class and take questions about how to think about the information given in the diagrams and how to write triangle congruence statements.
To change pace a little, but continue to focus on good thinking and reasoning, I plan to give students a writing assignment where they have to critique the reasoning of a fictional student, Barney. By asking students to critique Barney’s reasoning, students have to first wrap their head around Barney’s ideas, then come up with a way to convince Barney of the mistakes he has made so that he can learn from them. To justify their reasoning and explain at a high level, students must use precise academic and geometry vocabulary (MP3, MP6).
I like to collect these writing pieces without any discussion so that I can assess students’ understanding, which I will share out with them during the next lesson.
Resource Citation: I want to acknowledge the Geometry Team of Fremont High School in Sunnyvale, California, who came up with the idea of writing to our fictional student, Barney.
I want to give students time to synthesize their understanding of the triangle congruence shortcuts with important properties about special triangles and special quadrilaterals. In this station work, students can work individually or quietly with other students at their station; students can choose four of the five proofs to complete.
I expect students to begin thinking through their proofs by using appropriate symbols to mark the diagrams, which serves as evidence of their thinking. At the end of this station work, I have one student volunteer to display their proofs with the document camera so that everyone in the class may check their work.
I launch the idea of corresponding parts of congruent triangles being congruent by reviewing our class' ideas of congruence--this means going back to the 4-Triangles problem from the beginning of the year, where we talked about congruent shapes as being ones that "covered" one another completely. Then I ask my class to consider real-world, mass-produced items. I use whiteboard markers, asking students to consider that since they are mass-produced, they must be congruent. Then I ask them to consider the caps, the felt tips, the ink, which the students all agree are "congruent." What I want students to take away from this small real-world example is the idea that once we have established items to be congruent, we can conclude that all their corresponding parts are congruent.
I then do a whole-class example where I apply the idea of CPCTC in the proof so that students get a feeling for what is expected in this kind of proof.
In today's check for understanding, I want to assess students' proof writing and their understanding of how to use the idea that corresponding parts of congruent triangles are congruent. Students have to bring in their knowledge of the definition of a parallelogram to then prove that a pair of triangles are congruent to then show that the diagonals of the parallelogram bisect each other.