 Secondary Mathematics I

Strand: GEOMETRY - Congruence (G.CO)

Experiment with transformations in the plane. Build on student experience with rigid motions from earlier grades (Standards G.CO.1-5). Understand congruence in terms of rigid motions. Rigid motions are at the foundation of the definition of congruence. Reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems (Standards G.CO.6-8). Make geometric constructions (Standards G.CO.12-13).
• 3D Transmographer
This lesson contains an applet that allows students to explore translations, reflections, and rotations.
• Angle bisection and midpoints of line segments
This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment.
• 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.
• Bisecting an angle
This task provides the most famous construction to bisect a given angle.
• 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.
• 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.
• Congruence Criterion
The goal of this task is to establish the SSS congruence criterion using rigid motions.
• Construction of perpendicular bisector
The construction of the perpendicular bisector of a line segment is one of the most common in plane geometry and it is undertaken here.
• Constructions
This site provides both a video and step-by-step directions on how to complete a variety of constructions.
• Defining Parallel Lines
The goal of this task is to critically analyze several possible definitions for parallel lines.
• Defining Perpendicular Lines
The purpose of this task is to critically examine some different possible definitions of what it means for two lines to be perpendicular.
• Defining Reflections
The goal of this task is to compare and contrast the visual intuition we have of reflections with their technical mathematical definition.
• Defining Rotations
The goal of this task is to encourage students to be precise in their use of language when making mathematical definitions.
• Dilations and Distances
The goal of this task is to study the impact of dilations on distances between points.
• Evaluating Conditions for Congruency
This lesson unit is intended to help educators 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. They will also identify and understand the significance of a counter-example, and prove and evaluate proofs in a geometric context.
• Fixed points of rigid motions
The purpose of this task is to use fixed points at a tool for studying and classifying rigid motions of the plane.
• GEOMETRY - Congruence (G.CO) - Sec Math I Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics I - Congruence (G.CO).
• Geometry Construction Reference
Thirteen straightedge and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed.
• Geometry in Tessellations
In this lesson students will learn about lines, angles, planes, and experiment with the area and perimeter of polygons.
• Horizontal Stretch of the Plane
The goal of this task is to compare a transformation of the plane (translation) which preserves distances and angles to a transformation of the plane (horizontal stretch) which does not preserve either distances or angles.
• Identifying Rotations
The purpose of this task is to use the definition of rotations in order to find the center and angle of rotation given a triangle and its image under a rotation.
• Identifying Translations
The purpose of this task is to study the impact of translations on triangles.
• Identifying Unknown Transformations
This applet allows the student to drag a shape and then observe the changes to its behavior. They then determine whether the alteration is due to reflection, a rotation, or a translation/slide transformation.
• Inscribing a hexagon in a circle
This task is primarily for instructive purposes but can be used for assessment as well. Parts (a) and (b) are good applications of geometric constructions using a compass and could be used for assessment purposes but the process is a bit long since there are six triangles which need to be constructed.
• Inscribing a square in a circle
This task provides an opportunity for students to apply triangle congruence theorems in an explicit, interesting context.
• Inscribing an equilateral triangle in a circle
This task implements many important ideas from geometry including trigonometric ratios, important facts about triangles, and reflections. As a result, it is recommended that this task be undertaken relatively late in the geometry curriculum.
• Inscribing and Circumscribing Right Triangles
This lesson unit is intended to help educators assess how well students are able to use geometric properties to solve problems.
• Introduction to Constructions
Introduction to Euclidean Construction - tools and rules.
• Introduction to the Materials (Math 1)
Introduction to the Materials in the Mathematics One of the The MVP classroom experience begins by confronting students with an engaging task and then invites them to grapple with solving it. As students ideas emerge, take form, and are shared, the teacher orchestrates the student discussions and explorations towards a focused mathematical goal. As conjectures are made and explored, they evolve into mathematical concepts that the community of learners begins to embrace as effective strategies for analyzing and solving problems.
• Locating Warehouse
This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions,. Alternatively, this could be part of a full introduction to angle bisectors, culminating in a full proof that the three angle bisectors are concurrent, an essentially complete proof of which is found in the solution below.
• Module 6: Transformations & Symmetry - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 6, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection.
• Module 6: Transformations & Symmetry - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 6 Teacher Notes, Transformations and Symmetry, builds on students experiences with rigid motion in earlier grades to formalize the definitions of translation, rotation, and reflection.
• Module 7: Congruence, Construction & Proof - Student Edition (Math 1)
The Mathematics Vision Project, Secondary Math One Module 7, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles.
• Module 7: Congruence, Construction & Proof - Teacher Notes (Math 1)
The Mathematics Vision Project, Secondary Math One Module 7 Teacher Notes, Congruence, Construction, and Proof, begins by developing constructions as another tool to be used to reason about figures and to justify properties of shapes. Individual constructions are not taught for the sake of memorizing a series of steps, but rather to reason using known properties of shapes such as circles.
• Origami equilateral triangle
The purpose of this task is to explore reflections in the context of paper folding.
• Origami regular octagon
The goal of this task is to study the geometry of reflections in the context of paper folding.
• Patterns in Fractals
In this lesson students will be introduced to patterns, the terminology used in patterns, and practice finding patterns in the observable process of fractal generation.
• Placing a Fire Hydrant
This task can be implemented in a variety of ways. For a class with previous exposure to properties of perpendicular bisectors, part (a) could be a quick exercise in geometric constructions, and an application of the result. Alternatively, this could be part of an introduction to perpendicular bisectors, culminating in a full proof that the three perpendicular bisectors are concurrent at the circumcenter of the triangle.
• Properties of Congruent Triangles
The goal of this task is to understand how congruence of triangles, defined in terms of rigid motions, relates to the corresponding sides and angles of these triangles.
• Reflected Triangles
This task is a reasonably straight-forward application of rigid motion geometry, with emphasis on ruler and straightedge constructions, and would be suitable for assessment purposes.
• 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.
• Reflections and Equilateral Triangles II
This task examines some of the properties of reflections of the plane which preserve an equilateral triangle: these were introduced in ''Reflections and Isosceles Triangles'' and ''Reflection and Equilateral Triangles I''.
• Reflections and Isosceles 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 focussing on the class of isosceles triangles.
• Seven Circles II
This task is intended primarily for instructional purposes. It provides a concrete geometric setting in which to study rigid transformations of the plane
• Showing a triangle congruence: a particular case
This task provides experience working with transformations of the plane and also an abstract component analyzing the effects of the different transformations.
• Showing a triangle congruence: the general case
The purpose of this task is to work with transformations to exhibit triangle congruences in a general setting.
• Symmetries of a circle
This task asks students to examine lines of symmetry using the high school definition of reflections.
• Symmetries of a quadrilateral I
This task provides an opportunity to examine the taxonomy of quadrilaterals from the point of view of rigid motions.
• Symmetries of a quadrilateral II
This task examines quadrilaterals from the point of view of rigid motions and complements.
• Symmetries of rectangles
This task examines the rigid motions which map a rectangle onto itself.
• Taking a Spin (pdf)
Although students are often asked to find the angles of rotational symmetry for given regular polygons, in this task they are asked to find the regular polygons for a given angle of rotational symmetry, a reversal that yields some surprising results. This task would be most appropriate with students who have at least some experience in exploring rotational symmetry.
• Tangent Lines and the Radius of a Circle
This task presents a foundational result in geometry, presented with deliberately sparse guidance in order to allow a wide variety of approaches.
• Tessellations: Geometry and Symmetry
Students can explore polygons, symmetry, and the geometric properties of tessellations in this lesson.
• Translations, Reflections, and Rotations
Students are introduced to the concepts of translation, reflection and rotation in this lesson plan.
• Trigonometric Identities and Rigid Motions
The purpose of this task is to apply translations and reflections to the graphs of the equations f(x)=cosx and g(x)=sinx in order to derive some trigonometric identities.
• Unit Squares and Triangles
This problem provides an opportunity for a rich application of coordinate geometry.
• Visual Patterns in Tessellations
In this lesson students will learn about types of polygons and tessellation patterns around us.
• When Does SSA Work to Determine Triangle Congruence?
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. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent.
• Why Does ASA Work?
The two triangles in this problem share a side so that only one rigid transformation is required to exhibit the congruence between them. In general more transformations are required and the "Why does SSS work?'' and "Why does SAS work?'' problems show how this works.
• Why does SAS work?
For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections.
• Why does SSS work?
This particular sequence of transformations which exhibits a congruency between triangles ABC and DEF used one translation, one rotation, and one reflection. http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey  Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer  Throndsen.

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.