# Standard 2: Apply properties of transformations to describe congruence or similarity.

General Information
Number: MA.912.GR.2
Title: Apply properties of transformations to describe congruence or similarity.
Type: Standard
Subject: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning

## Related Benchmarks

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MA.912.GR.2.AP.1a
Given a preimage and image, identify the transformation.
MA.912.GR.2.AP.1b
Select the algebraic coordinates that represent the transformation.
MA.912.GR.2.AP.2
Select a transformation that preserves distance.
MA.912.GR.2.AP.3
Identify a given sequence of transformations, that includes translations or reflections, that will map a given figure onto itself or onto another congruent figure.
MA.912.GR.2.AP.5
Given a geometric figure and a sequence of transformations, select the transformed figure on a coordinate plane.
MA.912.GR.2.AP.6
Use rigid transformations that includes translations or reflections to map one figure onto another to show that the two figures are congruent.
MA.912.GR.2.AP.8
Identify an appropriate transformation to map one figure onto another to show that the two figures are similar.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## Formative Assessments

Justifying HL Congruence:

Students are asked to use rigid motion to explain why the HL pattern of congruence ensures right triangle congruence.

Type: Formative Assessment

Dilation of a Line: Factor of Two:

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Type: Formative Assessment

Dilation of a Line: Factor of One Half:

Students are asked to graph the image of three points on a line after a dilation using a center not on the line and to generalize about dilations of lines when the line does not contain the center.

Type: Formative Assessment

Dilation of a Line Segment:

Students are asked to dilate a line segment and describe the relationship between the original segment and its image.

Type: Formative Assessment

Congruent Trapezoids:

Students will determine whether two given trapezoids are congruent.

Type: Formative Assessment

Justifying a Proof of the AA Similarity Theorem:

Students are asked to justify statements of a proof of the AA Similarity Theorem.

Type: Formative Assessment

Prove the AA Similarity Theorem:

Students will indicate a complete proof of the AA Theorem for triangle similarity.

Type: Formative Assessment

Reflect a Semicircle:

Students are asked to reflect a semicircle across a given line.

Type: Formative Assessment

Dilation of a Line: Center on the Line:

Students are asked to graph the image of two points on a line after a dilation using a center on the line and to generalize about dilations of lines when the line contains the center.

Type: Formative Assessment

Two Triangles:

Students are asked to describe the transformations that take one triangle onto another.

Type: Formative Assessment

Transform This:

Students are asked to translate and rotate a triangle in the coordinate plane and explain why the pre-image and image are congruent.

Type: Formative Assessment

Type: Formative Assessment

Repeated Reflections and Rotations:

Students are asked to describe what happens to a triangle after repeated reflections and rotations.

Type: Formative Assessment

Indicate the Transformations:

Students are asked to describe the transformations that take one triangle onto another.

Type: Formative Assessment

Demonstrating Rotations:

Type: Formative Assessment

Similarity - 2:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Type: Formative Assessment

Similarity - 1:

Students are asked to describe a sequence of transformations to show that two polygons are similar.

Type: Formative Assessment

Similarity - 3:

Students are asked to describe a sequence of transformations that demonstrates two polygons are similar.

Type: Formative Assessment

Proving Similarity:

Students are asked to explain similarity in terms of transformations.

Type: Formative Assessment

Proving the Alternate Interior Angles Theorem:

In a diagram involving two parallel lines and a transversal, students are asked to use rigid motion to prove that alternate interior angles are congruent.

Type: Formative Assessment

Proving Congruence:

Students are asked to explain congruence in terms of rigid motions.

Type: Formative Assessment

Multistep Congruence:

Students are asked to describe a sequence of rigid motions to demonstrate the congruence of two polygons.

Type: Formative Assessment

Rigid Motion - 3:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Type: Formative Assessment

Rigid Motion - 2:

Students are asked to describe a rigid motion to demonstrate two polygons are congruent.

Type: Formative Assessment

Rigid Motion - 1:

Students are asked to describe a rigid motion to demonstrate that two polygons are congruent.

Type: Formative Assessment

Showing Triangles Congruent Using Rigid Motion:

Students are asked to use the definition of congruence in terms of rigid motion to show that two triangles are congruent in the coordinate plane.

Type: Formative Assessment

Proving Congruence Using Corresponding Parts:

Students are asked to prove two triangles congruent given that all pairs of corresponding sides and angles are congruent.

Type: Formative Assessment

Showing Similarity:

Students are asked to use the definition of similarity in terms of similarity transformations to determine whether or not two quadrilaterals are similar.

Type: Formative Assessment

The Consequences of Similarity:

Students are given the definition of similarity in terms of similarity transformations and are asked to explain how this definition ensures the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Type: Formative Assessment

To Be or Not To Be Similar:

Students are asked to use the definition of similarity in terms of similarity transformations to determine whether or not two triangles are similar.

Type: Formative Assessment

Justifying SSS Congruence:

Students are asked to use rigid motion to explain why the SSS pattern of congruence ensures triangle congruence.

Type: Formative Assessment

Justifying SAS Congruence:

Students are asked to use rigid motion to explain why the SAS pattern of congruence ensures triangle congruence.

Type: Formative Assessment

Justifying ASA Congruence:

Students are asked to use rigid motion to explain why the ASA pattern of congruence ensures triangle congruence.

Type: Formative Assessment

Showing Congruence Using Corresponding Parts - 2:

Students are given two triangles in which all pairs of corresponding parts are congruent and are asked to use the definition of congruence in terms of rigid motion to show the triangles are congruent.

Type: Formative Assessment

Showing Congruence Using Corresponding Parts - 1:

Students are given two triangles in which all pairs of corresponding parts are congruent and are asked to use the definition of congruence in terms of rigid motion to show the triangles are congruent.

Type: Formative Assessment

Congruence Implies Congruent Corresponding Parts:

Students are given two congruent triangles and asked to determine the corresponding side lengths and angle measures and to use the definition of congruence in terms of rigid motion to justify their reasoning.

Type: Formative Assessment

Transformations And Functions:

Students are given examples of three transformations and are asked if each is a function.

Type: Formative Assessment

Comparing Transformations:

Students are asked to determine whether or not dilations and reflections preserve distance and angle measure.

Type: Formative Assessment

Demonstrating Translations:

Students are asked to translate a quadrilateral according to a given vector.

Type: Formative Assessment

Demonstrating Reflections:

Type: Formative Assessment

## Lesson Plans

Transformations in the Coordinate Plane:

In this exploration activity of reflections, translations, and rotations, students are guided to discover general algebraic rules for special classes of transformations in the coordinate plane. This lesson is intended to be used after the development of formal definitions of rotations, translations, and reflections.

Type: Lesson Plan

"Triangle Congruence Show" Starring Rigid Transformations:

Students will be introduced to the definition of congruence in terms of rigid motion and use it to determine if two triangles are congruent.

Type: Lesson Plan

Reflecting on the Commute:

Students are given a set of coordinates that indicate a specific triangle on a coordinate plane. They will also be given a set of three reflections to move the triangle through. Students will then perform three other sequences of reflections to determine if the triangle ends up where it started.

Type: Lesson Plan

Transformation and Similarity:

Using non-rigid motions (dilations), students learn how to show that two polygons are similar. Students will write coordinate proofs confirming that two figures are similar.

Type: Lesson Plan

Sequence of Transformations:

This lesson will assist students in performing multi-step transformations. Students will follow a sequence of transformations on geometric figures using translations, reflections, and rotations.

Type: Lesson Plan

Regular Polygon Transformation Investigation:

This is an introductory lesson on regular polygon transformation for congruency using a hands-on approach.

Type: Lesson Plan

Match That!:

Students will prove that two figures are congruent based on a rigid motion(s) and then identify the corresponding parts using paragraph proof and vice versa, prove that two figures are congruent based on corresponding parts and then identify which rigid motion(s) map the images.

Type: Lesson Plan

Isometries with GeoGebra:

In this lesson, students explore transformations with GeoGebra and then apply concepts using a straightedge on paper. Students apply rules for each isometry. There is a teacher-driven opening followed by individual student activity.

Type: Lesson Plan

How Did It Get There? A Series of Transformation Events:

Students will perform a series of transformations in order to determine how the pre-image will map onto the final image of a given figure. Students will use patty paper to manipulate their pre-image onto the image. Students will also work in collaborative groups to discuss their findings and will have the opportunity to share their series of transformations with the class. The class discussion will be used to demonstrate that there are several ways for the students to map their pre-image onto the final image.

Type: Lesson Plan

Rotations of Regular Polygons:

This lesson guides students through the development of a formula to find the first angle of rotation of any regular polygon to map onto itself. Free rotation simulation tools such as GeoGebra, are used.

Type: Lesson Plan

Dancing For Joy:

We have danced our way through reflections, rotations, and translations; now we are ready to take it up a notch by performing a sequence of transformations. Students will also discover the results when reflecting over parallel lines versus intersecting lines.

Type: Lesson Plan

Turning to Congruence:

This lesson uses rigid motions to prove the ASA and HL triangle congruence theorems.

Type: Lesson Plan

Product of Two Transformations:

Students will identify a sequence of steps that will translate a pre-image to its image. Students will also demonstrate that the sequence of two transformations is not always commutative.

Type: Lesson Plan

Slip, Slide, Tip, and Turn: Corresponding Angles and Corresponding Sides:

Using the definition of congruence in terms of rigid motion, students will show that two triangles are congruent.

Type: Lesson Plan

Teach your students to maneuver a "spaceship" through a sequence of transformations that will successfully carry it onto its landing pad. GeoGebra directons are provided.

Type: Lesson Plan

Where Will I Land?:

This is a beginning level lesson on predicting the effect of a series of reflections or a quick review of reflections for high school students.

Type: Lesson Plan

Exploring Congruence Using Transformations:

This is an exploratory lesson that elicits the relationship between the corresponding sides and corresponding angles of two congruent triangles.

Type: Lesson Plan

How Much Proof Do We Need?:

Students determine the minimum amount of information needed to prove that two triangles are similar.

Type: Lesson Plan

How do your Air Jordans move?:

In this inquiry lesson, students are moving their individually designed Air Jordans around the room to explore rigid transformations on their shoes. They will Predict-Observe-Explain the transformations and then have to explain their successes/failures to other students.

Type: Lesson Plan

I Am Still Me Transformed.:

Students explore ways of applying, identifying, and describing reflection and rotation symmetry for both geometric and real-world objects, for them to develop a better understanding of symmetries in transformational geometry.

Type: Lesson Plan

Triangles on a Lattice:

In this activity, students will use a 3x3 square lattice to study transformations of triangles whose vertices are part of the lattice. The tasks include determining whether two triangles are congruent, which transformations connect two congruent triangles, and the number of non-congruent triangles (with vertices on the lattice) that are possible.

Type: Lesson Plan

Rotations and Reflections of an Equilateral Triangle:

Students will apply simple transformations (rotation and reflection) to an equilateral triangle, then determine the result of the action of two successive transformations, eventually determining whether the action satisfies the commutative and associate properties.

Type: Lesson Plan

Congruence vs. Similarity:

Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong.

Type: Lesson Plan

Transformations... Geometry in Motion:

Students will practice and compare transformations, and then determine which have isometry. Students should have a basic understanding of the rules for each transformation as they will apply these rules in this activity. There is a teacher-led portion in this lesson followed by partner activity. Students will be asked to explain and justify their reasoning,

S

Type: Lesson Plan

Reflections Hands On:

Students will use a protractor/ruler to construct reflections and a composite of reflections. They will create transformations using paper cut-outs and a coordinate plane. For independent practice, students will predict and verify sequences of transformations. The teacher will need an LCD Projector and document camera.

Type: Lesson Plan

Circle to Circle:

Students use coordinate-based translations and dilations to prove circles are similar.

Type: Lesson Plan

A Transformation's Adventure with Patty Paper: Exploring Translations, Reflections and Rotations.:

Students are introduced to isometric transformations using patty paper. Translations, reflections, and rotations will be explained and practiced, emphasizing the properties preserved during those transformations and, without sacrificing precision, allowing students to differentiate between these isometries. The lesson can also be taught using GeoGebra free software.

Type: Lesson Plan

Flipping Fours:

Students will translate, rotate and reflect quadrilaterals (Parallelogram, Rectangle, Square, Kite, Trapezoid, and Rhombus) using a coordinate grid created on the classroom floor and on graph paper. This activity should be used following guided lessons on transformations.

Type: Lesson Plan

Let's Reflect On This...:

Students will use parallel and intersecting lines on the coordinate plane to transform reflections into translations and rotations.

Type: Lesson Plan

Dilation Transformation:

Students identify dilations, verify that polygons are similar, and use the dilation rule to map dilations. Task cards are provided for independent practice. The PowerPoint also includes detailed illustrations for constructing a dilation using a compass and a straight edge.

Type: Lesson Plan

Transformers 3:

Students will learn the vocabulary of three rigid transformations, reflection, translation, and rotation, and how they relate to congruence. Students will practice transforming figures by applying each isometry and identifying which transformation was used on a figure. The teacher will assign students to take pictures of the three transformations found in the real world.

Type: Lesson Plan

Altitude to the Hypotenuse:

Students will discover what happens when the altitude to the hypotenuse of a right triangle is drawn. They learn that the two triangles created are similar to each other and to the original triangle. They will learn the definition of geometric mean and write, as well as solve, proportions that contain geometric means. All discovery, guided practice, and independent practice problems are based on the powerful altitude to the hypotenuse of a right triangle.

Type: Lesson Plan

Transform through the Maze:

In this fun activity, students will use rigid transformations to move a triangle through a maze. The activity provides applications for both honors and standard levels. It requires students to perform rotations, translations, and reflections.

Type: Lesson Plan

Polygon Transformers:

This guided discovery lesson introduces students to the concept that congruent polygons can be formed using a series of transformations (translations, rotations, reflections). As a culminating activity, students will create a robot out of transformed figures.

Type: Lesson Plan

Rotations and Reflections of an Equilateral Triangle:

Students will apply simple transformations (rotation and reflection) to an equilateral triangle, then determine the result of the action of two successive transformations, eventually determining whether the action satisfies the commutative and associate properties.

Type: Lesson Plan

## Original Student Tutorial

Home Transformations:

Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiasts

All Circles Are Similar- Especially Circular Pizza!:

What better way to demonstrate that all circles are similar then to use pizzas! Gaines Street Pies explains how all pizza pies are similar through transformations.

Type: Perspectives Video: Professional/Enthusiast

Reflections, Rotations, and Translations with Additive Printing:

<p>Transform your understanding of 3D modeling when you learn about how shapes are manipulated to arrive at a final 3D printed form!</p>

Type: Perspectives Video: Professional/Enthusiast

3D Modeling with 3D Shapes:

<p>Complex 3D shapes are often created using simple 3D primitives! Tune in and shape up as you learn about this application of geometry!</p>

Type: Perspectives Video: Professional/Enthusiast

Making Candy: Uniform Scaling:

<p>Don't be a shrinking violet. Learn how uniform scaling is important for candy production.</p>

Type: Perspectives Video: Professional/Enthusiast

Bacteriophage Geometry and Structure:

<p>Viruses aren't alive but they still need to stay in shape! Learn more about the geometric forms of bacteriophages!</p>

Type: Perspectives Video: Professional/Enthusiast

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Why Does ASA Work?:

This problem solving task ask students to show the reflection of one triangle maps to another triangle.

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Why Does SAS Work?:

This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points.

Reflections and Isosceles Triangles:

This activity uses rigid transformations of the plane to explore symmetries of classes of triangles.

Reflections and Equilateral Triangles II:

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Reflections and Equilateral Triangles:

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Why does SSS work?:

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Building a tile pattern by reflecting octagons:

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square.

Building a tile pattern by reflecting hexagons:

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Congruent Triangles:

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Triangle congruence with coordinates:

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

## Text Resource

Math for Hungry Birds:

This informational text resource is intended to support reading in the content area. A new study indicates that the flying patterns of hunting albatrosses may resemble mathematical designs called fractals. This article describes the basics of fractals and why scientists think the albatross may hunt in such patterns. As it turns out, many animals may use math to find food!

Type: Text Resource

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorial

Home Transformations:

Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

Making Candy: Uniform Scaling:

<p>Don't be a shrinking violet. Learn how uniform scaling is important for candy production.</p>

Type: Perspectives Video: Professional/Enthusiast

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Why Does ASA Work?:

This problem solving task ask students to show the reflection of one triangle maps to another triangle.

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Why Does SAS Work?:

This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points.

Reflections and Isosceles Triangles:

This activity uses rigid transformations of the plane to explore symmetries of classes of triangles.

Reflections and Equilateral Triangles II:

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Reflections and Equilateral Triangles:

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Why does SSS work?:

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Building a tile pattern by reflecting octagons:

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square.

Building a tile pattern by reflecting hexagons:

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Congruent Triangles:

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

Triangle congruence with coordinates:

In this resource, students will decide how to use transformations in the coordinate plane to translate a triangle onto a congruent triangle. Exploratory examples are included to prompt analytical thinking.

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

## Perspectives Video: Professional/Enthusiast

Making Candy: Uniform Scaling:

<p>Don't be a shrinking violet. Learn how uniform scaling is important for candy production.</p>

Type: Perspectives Video: Professional/Enthusiast

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Partitioning a Hexagon:

The purpose of this task is for students to find a way to decompose a regular hexagon into congruent figures. This is meant as an instructional task that gives students some practice working with transformations.

Why Does ASA Work?:

This problem solving task ask students to show the reflection of one triangle maps to another triangle.

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Why Does SAS Work?:

This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points.

Reflections and Isosceles Triangles:

This activity uses rigid transformations of the plane to explore symmetries of classes of triangles.

Reflections and Equilateral Triangles II:

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Reflections and Equilateral Triangles:

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Why does SSS work?:

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Building a tile pattern by reflecting octagons:

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square.

Building a tile pattern by reflecting hexagons:

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Congruent Segments:

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Congruent Triangles:

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''