Lesson 4 | Congruence in Two Dimensions | 10th Grade Mathematics

Objective

Determine congruence of two dimensional figures by translation.

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.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.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

Describe how translation of points and line segments can be extended to translate polygonal figures.
Translate polygons described by algebraic rules (specifically triangles, rectangles, parallelograms, and regular polygons) on the coordinate plane.
Translate polygons described by vectors (specifically triangles, rectangles, parallelograms, and regular polygons) off the coordinate plane using constructions and patty paper.
Determine whether two polygons are congruent through transformation by translation.
Describe why two polygons are congruent and highlight the angle and distance preservation of translations as evidence.

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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 are four original points, and four points after transformation.

Are polygon $${ABCD}$$ and polygon $${A'B'C'D'}$$ congruent? Explain your reasoning.

Guiding Questions

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Problem 2

Allison says that the following two figures are congruent because they have the “same size, same shape”. Describe why her reasoning is incorrect, using the properties of translation as evidence.

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

Create a congruent figure to the one shown in the diagram below.

Identify the translation rule you used for the figure.
Describe how you know the figures are congruent using properties of translations.

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:
- Trace figure on patty paper and translate according to an algebraic rule.
- Given a transformation, identify algebraic rule.
- Translate one point of a geometric figure; identify the rest of the figure and the algebraic rule.
- Identification of reflection, rotation, and translation.
- Question about relationship between line segments of translated figure. (Draw a line between corresponding points between translated figures.) What is the relationship between all of the lines drawn?

Desmos Polygraph: Translations
Illustrative Mathematics Identifying Translations

Lesson 3

Lesson 5

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

Target Task

Additional Practice

Lesson Map

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