# Cluster 2: Understand congruence in terms of rigid motions. (Geometry - Major Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.G-CO.2
Title: Understand congruence in terms of rigid motions. (Geometry - Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Geometry: Congruence

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.G-CO.2.AP.7a
Use definitions to demonstrate congruency and similarity in figures.
MAFS.912.G-CO.2.AP.6a
Use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on figures in the coordinate plane.
MAFS.912.G-CO.2.AP.6b
Knowing that rigid transformations preserve size and shape or distance and angle, use this fact to connect the idea of congruency and develop the definition of congruent.
MAFS.912.G-CO.2.AP.8a
Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria; ASA, SSS, and SAS.

## Related Resources

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

## Educational Software / Tool

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

## 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

Congruent Trapezoids:

Students will determine whether two given trapezoids are congruent.

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

Repeated Reflections and Rotations:

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

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

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

## Lesson Plans

Coding Geometry Challenge #23 & 24:

This set of geometry challenges focuses on using transformations to show similarity and congruence of polygons and circles. Students problem solve and think as they learn to code using block coding software.  Student will need to use their knowledge of the attributes of polygons and mathematical principals of geometry to accomplish the given challenges. The challenges start out fairly simple and move to more complex situations in which students can explore at their own pace or work as a team. Computer Science standards are seamlessly intertwined with the math standards while providing “Step it up!” and “Jump it up!” opportunities to increase rigor.

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

Turning to Congruence:

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

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

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 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

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

Transformers 3:

Lesson is introduced by teacher activating students' prior knowledge of congruence. The teacher will present the power point that defines and illustrates transformations by rigid motion. The students will answer questions on their worksheets at the time indicated in the power point. The students will work on their independent assignment. The teacher will give a short quiz to test for understanding. The teacher will assign for the students to take pictures of the three transformations found in the real world.

Type: Lesson Plan

Analyzing Congruence Proofs:

This lesson is intended to help you assess how well students are able to:

• Work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles.
• Identify and understand the significance of a counter-example.
• Prove, and evaluate proofs in a geometric context.

Type: Lesson Plan

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.

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

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.

Are the Triangles Congruent?:

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

## Tutorials

Finding congruent triangles:

In this tutorial, students will use the SSS, ASA, SAS, and AAS postulates to find congruent triangles

Type: Tutorial

Using SSS in a proof:

This tutorial discusses the difference between a theorem and axiom. It also shows how to use SSS in a proof.

Type: Tutorial

Triangle congruence postulates:

This tutorial discusses SSS, SAS, ASA and AAS postulates for congruent triangles. It also shows AAA is only good for similarity and SSA is good for neither.

Type: Tutorial

Congruent Triangles and SSS:

In this video, students will learn about congruent triangles and the "Side-Side-Side" postulate.

Type: Tutorial

## Worksheet

Students are provided two pairs of quadrilaterals with information that relates those in each of the pairs. The objective is to prove that each of the pairs are congruent. The teacher will provide guidance to the students as necessary by suggesting a triangle congruence theorem and/or construction of a diagonal.

Type: Worksheet

## Student Resources

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

## Educational Software / Tool

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

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.

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

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.

Are the Triangles Congruent?:

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

## Tutorials

Finding congruent triangles:

In this tutorial, students will use the SSS, ASA, SAS, and AAS postulates to find congruent triangles

Type: Tutorial

Using SSS in a proof:

This tutorial discusses the difference between a theorem and axiom. It also shows how to use SSS in a proof.

Type: Tutorial

Triangle congruence postulates:

This tutorial discusses SSS, SAS, ASA and AAS postulates for congruent triangles. It also shows AAA is only good for similarity and SSA is good for neither.

Type: Tutorial

Congruent Triangles and SSS:

In this video, students will learn about congruent triangles and the "Side-Side-Side" postulate.

Type: Tutorial

## Parent Resources

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

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.

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

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.