In this unit, students prove a variety of figures congruent. They start with segments (2 vertices), then move on to triangles (3 vertices). Before starting this unit, students are familiar with rigid transformations and congruence from a previous unit. The triangle congruence theorems in this unit lay the foundation for triangle similarity in a later course.

The first section focuses on establishing congruence. Students recall from middle school that if two figures are congruent, then each pair of corresponding parts of the figures are also congruent. Next, students use rigid transformations to justify the triangle congruence theorems of Euclidean geometry: Side-Side-Side Triangle Congruence, Side-Angle-Side Triangle Congruence, and Angle-Side-Angle Triangle Congruence. Students justify that for each set of criteria, a sequence of rigid motions exists that will take one triangle onto the other. 

In a previous unit, students began justifying their responses. In this unit, students write more rigorous proofs. At first, students may use imprecise language to convey their ideas. Throughout the unit they will read examples, practice explaining ideas to a partner, and build a reference of precise statements to use in future proofs. Writing proofs doesn’t only mean providing the reasons for someone else’s claim, constructing a viable argument includes writing conjectures and detailing given information. Students write congruence proofs using transformations so the resulting theorems can be used in future work without repeating the argument.

The proofs in these materials are all written in narrative form. The narrative format matches the discussion students might have to use to convince their partner, and it also matches the way mathematicians write proofs. While students may use other formats to support their organization, it is important that students can see the flow of reasoning that exists in a well-written proof. A two-column proof can be thought of like an outline for an essay. Outlines help organize thoughts, but an outline is less persuasive than a well-written essay. Students should learn to write a well-written justification in the form of a narrative proof.

Students are learning ways to express their reasoning more formally. Expect students to put together explanations with various levels of formality. Mastery of proof writing is not expected by the end of this unit. Students will have more practice writing proofs in a later course.

Note on materials: For most activities in this unit, students have access to a geometry toolkit that includes many tools that students can choose from strategically: compass and straightedge, tracing paper, colored pencils, and scissors. In some lessons, students will also need access to a ruler and protractor. Finally, there are some activities that are best done using dynamic geometry software, and these lessons indicate that digital materials are preferred. Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.