Students identify, perform, and algebraically describe rigid motions to establish congruence of two dimensional polygons, including triangles, and develop congruence criteria for triangles.
In Unit 2, Congruence in Two Dimensions, students build off the work done in unit 1 to use rigid motions to establish congruence of two dimensional polygons, including triangles. Through a transformational lens, students develop congruence criteria for triangles that serve as the foundation for establishing relationships for right triangles and trigonometry in Unit 4. In this unit, students are also introduced to relationships with parallelograms, and this serves as a foundation for further polygon relationships in Unit 5.
In this unit, students will identify, perform, and describe transformations algebraically, using constructions both on and off the coordinate plane to justify congruence. Students will describe how the constructions used to perform the rigid motions point to the properties of the transformations, and, in some cases, indicate other transformations that are equivalent. In 8th grade, students worked with congruence through transformational geometry and learned about the interior angle sum theorem for triangles as well as the relationship between interior and exterior angles of a triangle. This understanding is formalized in this unit.
Students will learn about the criteria needed for triangle congruence, building from 7th grade, where math students learned to identify conditions that determine a unique triangle, more than one triangle, or no triangle. Students will also learn how and why these criteria exist by exploring the transformations underlying the criteria.
Because triangles make up most other polygons, examining and proving theorems about triangle congruence will help students develop the foundation needed to understand the properties and criteria for congruence of other polygons. The work of this unit is also particularly important preparation for trigonometry because trigonometry is based on triangle relationships. Developing an understanding of triangle congruence will ease students’ transition to trigonometry.
Pacing: 20 instructional days (18 lessons, 1 flex day, 1 assessment day)
This assessment accompanies Unit 2 and should be given on the suggested assessment day or after completing the unit.
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
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polygon | parallel/perpendicular | interior angles | exterior angles |
parallelogram | convex/concave | congruence | Perpendicular Bisector Theorem |
Hypotenuse Leg Congruence Criteria | property of opposite angles in parallelograms | Rotational Symmetry | Reflectional Symmetry |
Side Angle Side Congruence Criteria | Angle Side Angle Congruence Criteria | Side | Side Side Congruence Criteria |
Angle Angle Side Congruence Criteria | Base Angle of Isoceles Triangles Theorem | property of opposite sides in parallelograms | properties of diagonals in parallelograms |
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G.CO.A.1
G.CO.C.11
Define polygon and identify properties of polygons.
G.CO.C.10
Prove interior and exterior angle relationships in triangles.
G.CO.C.11
Describe and apply the sum of interior and exterior angles of polygons.
G.CO.A.2
G.CO.A.4
G.CO.A.5
G.CO.B.7
Determine congruence of two dimensional figures by translation.
G.CO.A.2
G.CO.A.3
G.CO.A.5
G.CO.B.7
Reflect two dimensional figures on and off the coordinate plane.
G.CO.A.2
G.CO.A.3
G.CO.A.5
G.CO.B.7
Rotate two dimensional figures on and off the coordinate plane.
G.CO.A.5
G.CO.B.6
Describe a sequence of rigid motions that map a pre-image to an image (specifically triangles, rectangles, parallelograms, and regular polygons).
G.CO.A.3
G.CO.A.5
G.CO.B.6
Describe single rigid motions, or sequences of rigid motions that have the same effect on a figure.
G.CO.B.7
G.CO.B.8
Develop the Side Angle Side criteria for congruent triangles through rigid motions.
G.SRT.B.5
Prove angle relationships using the Side Angle Side criteria.
G.CO.A.2
G.CO.C.9
G.CO.C.10
Prove and apply that the points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
G.CO.B.7
G.CO.B.8
Develop Angle, Side, Angle (ASA) and Side, Side, Side (SSS) congruence criteria. Describe how the criteria develop from rigid motions.
G.SRT.B.5
Use triangle congruence criteria, rigid motions, and other properties of lines and angles to prove congruence between different triangles.
G.CO.B.7
G.CO.B.8
G.CO.C.10
Prove triangles congruent using Angle, Angle, Side (AAS), and describe why AAA is not a congruency criteria.
G.CO.B.7
G.CO.C.10
Develop the Hypotenuse- Leg (HL) criteria, and describe the features of a triangle that are necessary to use the HL criteria.
G.CO.C.9
G.SRT.B.5
Use criteria for triangle congruence to prove relationships among angles and sides in geometric problems.
Key: Major Cluster Supporting Cluster Additional Cluster
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