In this section students create conjectures about angle relationships and prove them using what they know about rigid transformations. Students review rigid transformations to support this skill.

The primary work is on proof-writing skills. In order to focus on justification, many proofs are for ideas students were first introduced to in previous grades, such as supplementary, complementary, vertical, and adjacent angles. Students also prove the Triangle Angle Sum Theorem using the rigorous definitions reviewed and developed throughout this section.

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 section. A more reasonable goal is for students to consistently label and mark figures to indicate congruence which helps them communicate more precisely.

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.

This section intentionally allows extra time for students to learn new routines and establish norms for the year.

In this section, students explore properties of triangles. The first three lessons of this section are purposefully paced to allow class time to learn routines and build math community.

First, students make observations about possible side lengths of a triangle and practice communicating their ideas. Next, students write the Triangle Inequality Theorem as a conjecture, an initial step to the proof writing students will do in this unit. In the following lesson, students write the Triangle Inequality Theorem as an explicit mathematical statement, informally justify their ideas, and practice using the theorem in a problem in which they engage with aspects of mathematical modeling.

In the next three lessons, students explore what happens to sides in a triangle as the angle opposite that side changes. This leads to an understanding that longer sides are across from larger angles and vice versa. Students review angle pairs and triangle properties to solve for unknown angles in figures made up of several triangles and use those angles to make conclusions about the sides of the triangles. Students end the section with an Information Gap designed to solidify their understanding of triangle properties and to build their communication skills.

In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.

In this section students apply the triangle congruence theorems to write more proofs. They consider special quadrilaterals, the diagonals of quadrilaterals, and the triangles formed by those diagonals. Students have the opportunity to engage in all aspects of the proof process: generating conjectures, detailing specific statements to be proved, writing a proof, and critiquing a proof. They also encounter the challenge of overlapping triangles.