Math / 10th Grade / Unit 2: Congruence in Two Dimensions

Congruence in Two Dimensions

Students identify, perform, and algebraically describe rigid motions to establish congruence of two dimensional polygons, including triangles, and develop congruence criteria for triangles.

Math

Unit 2

10th Grade

Unit Summary

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)

Assessment

This assessment accompanies Unit 2 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that align to each question
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) to which the assessment points

Internalization of Trajectory of Unit

Read and annotate “Unit Summary."
Notice the progression of concepts through the unit using “Unit at a Glance”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

Rigid motions can be performed to map one figure to another in single or a composition of transformations. These transformations can be performed using constructions, patty paper, and on a coordinate plane using properties of perpendicular and parallel lines.
Congruence can be determined through identification of rigid motions that will map one figure onto another. Because rigid motions are angle and distance preserving, corresponding parts of transformed figures through rigid motions are congruent.
Geometric facts are established through proof by determining a sequence of logical statements and reasons that drive towards a conclusion that is the theorem proved. Through this process, we can establish new facts.
Triangle congruence can be proven using side and angle relationships between corresponding parts as well as rigid motions.

Vocabulary

Terms and notation that students learn or use in the unit

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

Materials

The materials, representations, and tools teachers and students will need for this unit

compass
straightedge

Lesson Map

Topic A: Introduction to Polygons

Define polygon and identify properties of polygons.

G.CO.A.1 G.CO.C.11

Prove interior and exterior angle relationships in triangles.

G.CO.C.10

Describe and apply the sum of interior and exterior angles of polygons.

G.CO.C.11

Topic B: Rigid Motion Congruence of Two-Dimensional Figures

Determine congruence of two dimensional figures by translation.

G.CO.A.2 G.CO.A.4 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.2 G.CO.A.3 G.CO.A.5 G.CO.B.7

Describe a sequence of rigid motions that map a pre-image to an image (specifically triangles, rectangles, parallelograms, and regular polygons).

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.A.3 G.CO.A.5 G.CO.B.6

Topic C: Triangle Congruence

Develop the Side Angle Side criteria for congruent triangles through rigid motions.

G.CO.B.7 G.CO.B.8

Prove angle relationships using the Side Angle Side criteria.

G.SRT.B.5

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.A.2 G.CO.C.10 G.CO.C.9

Develop Angle, Side, Angle (ASA) and Side, Side, Side (SSS) congruence criteria. Describe how the criteria develop from rigid motions.

G.CO.B.7 G.CO.B.8

Use triangle congruence criteria, rigid motions, and other properties of lines and angles to prove congruence between different triangles.

G.SRT.B.5

Prove triangles congruent using Angle, Angle, Side (AAS), and describe why AAA is not a congruency criteria.

G.CO.B.7 G.CO.B.8 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.B.7 G.CO.C.10

Use criteria for triangle congruence to prove relationships among angles and sides in geometric problems.

G.CO.C.9 G.SRT.B.5

Topic D: Parallelogram Properties from Triangle Congruence

Prove that the opposite sides and opposite angles of a parallelogram are congruent.

G.CO.C.11

Prove theorems about the diagonals of parallelograms.

G.CO.C.11

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Congruence

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Congruence

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

Congruence

G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
G.CO.A.3 — Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Congruence

G.CO.A.3 — Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G.CO.A.4 — Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Congruence

G.CO.A.4 — Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Congruence

G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
G.CO.B.6 — Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Congruence

G.CO.B.6 — Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G.CO.B.7 — Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Congruence

G.CO.B.7 — Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.B.8 — Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Congruence

G.CO.B.8 — Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Congruence

G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Congruence

G.CO.C.11 — Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
G.CO.C.9 — Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Congruence

G.CO.C.9 — Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Similarity, Right Triangles, and Trigonometry

G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Similarity, Right Triangles, and Trigonometry

G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Geometry

7.G.A.2

Geometry

7.G.A.2 — Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
8.G.A.1

Geometry

8.G.A.1 — Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.2

Geometry

8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3

Geometry

8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.A.5

Geometry

8.G.A.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.G.B.6

Geometry

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.

Future Standards

Standards in future grades or units that connect to the content in this unit

Similarity, Right Triangles, and Trigonometry

G.SRT.A.1

Similarity, Right Triangles, and Trigonometry

G.SRT.A.1 — Verify experimentally the properties of dilations given by a center and a scale factor:
G.SRT.A.2

Similarity, Right Triangles, and Trigonometry

G.SRT.A.2 — Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G.SRT.A.3

Similarity, Right Triangles, and Trigonometry

G.SRT.A.3 — Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G.SRT.B.4

Similarity, Right Triangles, and Trigonometry

G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.B.5

Similarity, Right Triangles, and Trigonometry

G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.SRT.C.6

Similarity, Right Triangles, and Trigonometry

G.SRT.C.6 — Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Unit 1

Constructions, Proof, and Rigid Motion

Unit 3

Dilations and Similarity