In this unit, 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.” The work gives them practice working with rational numbers and equations for angle relationships. Students analyze and describe cross-sections of prisms, pyramids, and polyhedra. They understand and use the formula for the volume of a right rectangular prism and solve problems involving area, surface area, and volume. Students should have access to their geometry toolkits so that they have an opportunity to select and use appropriate tools strategically.

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).

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as critiquing, explaining, interpreting, 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:

Critique

• Reasoning about measuring angles (Lesson 1).
• Reasoning about decomposition of prisms (Lesson 13).
• Reasoning about surface area of prisms (Lesson 14).

Explain

• How to measure angles (Lesson 2).
• How to find unknown angle measurements (Lessons 4 and 5).
• How to find the volume of prisms (Lessons 12 and 13).
• How to find the surface area of prisms (Lesson 14).

Interpret

• Situations involving intersecting lines in order to form a conjecture (Lesson 3).
• Which information is relevant to answer questions (Lesson 4).
• Equations representing angle measurements (Lesson 5).
• Situations involving volume and surface area (Lesson 15 and 16).

Justify

• Whether or not shapes are identical copies (Lesson 6).
• Whether or not measurements determine identical copies (Lesson 9).
• Whether or not measurements determine unique triangles (Lesson 10).

In addition, students are expected to use language to compare angle measurements, compare triangles in a set, compare cross-sections of figures, describe characteristics of pattern blocks, describe positioning and movement of side lengths and angles, and describe cross-sections of prisms and pyramids. Students also have opportunities to generalize about patterns of angle measurements, about categories for unique triangles, and about categories for cross-sections.

The table shows lessons where new terminology is first introduced in this course, 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 that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson	new terminology
	receptive	productive
7.7.1	straight angle adjacent angles degree	right angle
7.7.2	supplementary complementary angle measure protractor	measurement error degrees
7.7.3	vertical angles intersect vertex (of an angle)
7.7.4		supplementary vertical angles
7.7.5	perpendicular	complementary
7.7.6	identical copy condition	angle measurement side length quadrilateral
7.7.7	compass different triangle	intersect identical copy segment
7.7.8		condition different triangle
7.7.9	unique triangle parallel
7.7.10		protractor compass
7.7.11	cross-section base (of a prism or pyramid) vertex (of a pyramid) face	prism pyramid perpendicular parallel
7.7.12		volume cross-section base (of a prism or pyramid)
7.7.14		face perimeter
7.7.15		surface area