Lesson 6 | Congruence in Two Dimensions | 10th Grade Mathematics

Objective

Rotate two dimensional figures on and off the coordinate plane.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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.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.

Foundational Standards

The foundational standards covered in this lesson

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.

Criteria for Success

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

Rotate polygons described by algebraic rules (specifically triangles, rectangles, parallelograms, and regular polygons) on the coordinate plane.
Rotate polygons described by degree and direction (specifically triangles, rectangles, parallelograms, and regular polygons) off the coordinate plane using constructions and patty paper.
Determine whether two polygons (specifically triangles) are congruent through transformation by rotation.
Describe why two polygons are congruent and highlight the angle and distance preservation of rotations as evidence.
Given a polygon with rotational symmetry, describe why the polygon will map onto itself.
Identify all degrees of rotational symmetry in a regular polygon (order of rotational symmetry).
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.

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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

Below is triangle $${\triangle ABC}$$. Rotate this shape five times clockwise about the origin.

Guiding Questions

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Problem 2

Which one doesn't belong? Give a reason why each of the quadrants does not belong with the rest.

Guiding Questions

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References

Which One Doesn't Belong Shapes — Shape 36 from Daniel Ruiz Aguilera

Problem 3

Suppose $${{{ABCD}}}$$ is a quadrilateral for which there is exactly one rotation, through an angle larger than 0 degrees and less than 360 degrees, which maps to itself. Further, no reflections map $${{{ABCD}}}$$ to itself.

What shape is $${{{ABCD}}}$$?

Guiding Questions

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References

Illustrative Mathematics Symmetries of a Quadrilateral I

Symmetries of a Quadrilateral I, accessed on Aug. 10, 2017, 11:11 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Target Task

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

Rotate the following figure $${90^{\circ}}$$ counterclockwise about the origin, three times.

A shape is formed through these rotations, with center at the origin.
Does this shape have rotational symmetry? If so, by what order/degree?
Doe this shape have reflectional symmetry? If so, what is the line of symmetry? If not, describe why not.

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.

Include problems such as:
- Ask students to identify playing cards with rotational and reflectional symmetry.
- Ask students to find the center of a rectangle and then name the rotational and reflectional symmetry of the rectangle.
- Ask students to identify whether figures have 90°, 180°, or 270° rotational symmetry.
- Rotated image and pre-image are given with a center of rotation, off a coordinate plane, and students need to identify the angle and direction of rotation.
- Students need to rotate figures around the origin using the coordinate plane and also by using constructions.

Illustrative Mathematics Symmetries of rectangles
Better Lesson Rigid Motions and Congruence
Which One Doesn't Belong Shapes — Shape 37 from Simon Gregg
Illustrative Mathematics Sum of angles in a triangle
Illustrative Mathematics Symmetries of a Quadrilateral II
EngageNY Mathematics Geometry > Module 1 > Topic C > Lesson 15 — Problem Set

Lesson 5

Lesson 7

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

Congruence

Foundational Standards

Geometry

Geometry

Geometry

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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