Lesson 8 | Congruence in Two Dimensions | 10th Grade Mathematics

Objective

Describe single rigid motions, or sequences of rigid motions that have the same effect on a figure.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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.

Foundational Standards

The foundational standards covered in this lesson

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe how reflections/rotations of 180 degrees are related and translations/reflections are related when line segments are parallel to a line of reflection.
Relate the rotational symmetry of a figure to the reflectional symmetry of a figure and describe how the congruence of these two rigid motions applies to both reflectional and rotational symmetry.
Describe the features of a figure where some rigid motions have the same effect and some do not (ie: angles and line segments may not have the same set of congruent rigid motions).
Explain the characteristics of each transformation in the sequence that mimic a single transformation.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Describe whether the following statement is always, sometimes, or never true:
“If you reflect a figure across two parallel lines, the result can be described with a single translation rule.”

Guiding Questions

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Problem 2

Describe whether the following statement is always, sometimes, or never true:
“The reflection of a figure over two unique lines of reflection can be described by a rotation.”

Guiding Questions

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Problem 3

Johnny says three rotations of $${90^{\circ}}$$ about the center of the figure is the same as three reflections with lines that pass through the center, so a figure with order 4 rotational symmetry results in a figure that also has reflectional symmetry.

Describe, using evidence from the two drawings below, to support or refute Johnny’s statement.

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Describe whether the converse of the statement in Anchor Problem #2 is always, sometimes, or never true:

Converse: “The rotation of a figure can be described by a reflection of a figure over two unique lines of reflection.”

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Independent practice for this lesson will ensure that the criteria for sequences of transformations and parameters for transformations are met.

Illustrative Mathematics Building a tile pattern by reflecting hexagons
New Visions for Public Schools Unit 1 Big Idea 3 — "Equivalent Transformations" from Instructional Routine: Connecting Representations
Illustrative Mathematics Building a tile pattern by reflecting oxagons

Lesson 7

Lesson 9

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

Congruence in Two Dimensions

Objective

Common Core Standards

Core Standards

Congruence

Congruence

Congruence

Foundational Standards

Geometry

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Additional Practice

Lesson Map

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