In this unit, students explore translations, rotations, and reflections of plane figures in order to understand the structure of rigid transformations. They use the properties of rigid transformations to formally define what it means for shapes to be congruent.

Two triangles D E F and A B C on a coordinate plane. Triangle D E F is the image of triangles A B C after rotation of 90 degrees, followed by a translation left 3 and down 2 units. Triangle A B C has the coordinates A(2 comma negative 2), B(6 comma 0) and C(6 comma 2). Triangle D E F has the coordinates D(negative 1 comma 0), E(negative 3 comma 4) and F(negative 5 comma 4).

In earlier grades, students studied geometric measurement to find angle measures and side lengths of two-dimensional figures as well as applied area and perimeter formulas for polygons including rectangles, parallelograms, and triangles. In this unit, students build on this work as they identify corresponding congruent angles and side lengths of figures and their images under rigid transformations. In an upcoming unit, students will explore dilations and similar figures in the plane.

In the first section, students begin with an informal exploration of transformations in the plane, then increase their precision of language to describe translations, rotations, and reflections with formal descriptions, including coordinates.

Then students identify corresponding parts of figures and conclude that angles and distances are preserved under rigid transformations. Students use this property to reason about plane figures, including parallel lines cut by a transversal.

Students then learn the formal definition of "congruent" and use this definition to show that corresponding parts of congruent figures are also congruent. Students apply their understanding of congruence and rigid motions to justify that the sum of the interior angles in a triangle must be .

Students investigate whether sets of angle and side length measurements determine unique triangles or multiple triangles, or fail to determine triangles. Students also study and apply angle relationships, learning to understand and use the terms “complementary,” “supplementary,” “vertical angles,” and “unique.”

Note: It is not expected that students memorize which conditions result in a unique triangle, an impossible-to-create triangle, or multiple possible triangles. Understanding that, for example, side-side-side (SSS) information results in zero or exactly one triangle will be explored in high school geometry. At this level, students should attempt to draw triangles with the given information and notice that there is only one way to do it (or that it is impossible to do). In this unit, students reason about congruence and justify properties of figures using rigid transformations, but they are not required to create a formal proof. They will prove these and other geometric properties more formally in later courses.

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

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as describing, generalizing, and justifying. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Describe

• Movements of figures (Lessons 1 and 2).
• Observations about transforming parallel lines (Lesson 8).
• Transformations using corresponding points, line segments, and angles (Lesson 9).
• Observations about angle measurements (Lesson 14).
• Transformations found in tessellations and in designs with rotational symmetry (Lesson 18).

Generalize

• About categories for movement (Lesson 2).
• About rotating line segments (Lesson 7).
• About the relationship between vertical angles (Lesson 8).
• About transformations and congruence (Lesson 11).
• About corresponding segments and length (Lesson 11).
• About alternate interior angles (Lesson 12).
• About the sum of angles in a triangle (Lesson 14).
• About categories for unique triangles (Lesson 16).

Justify

• Whether or not rigid transformations could produce an image (Lesson 6).
• Whether or not shapes are congruent (Lesson 10).
• Whether or not polygons are congruent (Lesson 11).
• Whether or not triangles can be created from given angle measurements (Lesson 13).
• Whether or not measurements determine unique triangles (Lesson 17).

In addition, students are expected to explain and interpret directions for transforming figures and apply transformations to find specific images. Students are also asked to use language to compare rotations of a line segment and compare perimeters and areas of rectangles. Over the course of the unit, teachers can support students’ mathematical understandings by amplifying (not simplifying) language used for all of these purposes as students demonstrate and develop ideas.

The table shows lessons where new terminology is first introduced, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms from the Glossary appear bolded. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.