
3D Transmographer
This lesson contains an applet that allows students to explore translations, reflections, and rotations.

Congruence Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for the Secondary Mathematics 1  Congruence (G.CO).

Geometry in Tessellations
In this lesson students will learn about lines, angles, planes, and experiment with the area and perimeter of polygons.

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.

QuadrilateralsBeyond Definition  A Practice Understanding Task
Making and justifying properties of quadrilaterals using symmetry transformations (G.CO.3, G.CO.4, G.CO.6) Congruence, Construction and Proof Module  Task 7

QuadrilateralsBeyond Definition  Teacher Notes
Teacher notes for "QuadrilateralsBeyond Definition  A Practice Understanding" task.
(G.CO.3, G.CO.4, G.CO.6) Congruence, Construction and Proof Module  Task 7

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.

Symmetries of Quadrilaterals  A Develop Understanding Task
Finding rotational symmetry and lines of symmetry in special types of quadrilaterals (G.CO.3, G.CO.6) Congruence, Construction and Proof Module  Task 5

Symmetries of Quadrilaterals  Teacher Notes
Teacher notes for the "Symmetries of Quadrilaterals  A Develop Understanding" task
(G.CO.3, G.CO.6) Congruence, Construction and Proof Module  Task 5

Symmetries of Regular Polygons  A Solidify Understanding Task
Examining characteristics of regular polygons that emerge from rotational symmetry and lines of symmetry (G.CO.3, G.CO.6) Congruence, Construction and Proof Module  Task 6

Symmetries of Regular Polygons  Teacher Notes
Teacher notes for "Symmetries of Regular Polygons  A Solidify Understanding" task. (G.CO.3, G.CO.6) Congruence, Construction and Proof Module  Task 6

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.

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.
