# Ch 1: Common Core HS Geometry: Congruence

### About This Chapter

**Standard:** Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (CCSS.Math.Content.HSG-CO.A.1)

**Standard:** 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). (CCSS.Math.Content.HSG-CO.A.2)

**Standard:** Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. (CCSS.Math.Content.HSG-CO.A.3)

**Standard:** Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. (CCSS.Math.Content.HSG-CO.A.4)

**Standard:** 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. (CCSS.Math.Content.HSG-CO.A.5)

**Standard:** Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. (CCSS.Math.Content.HSG-CO.A.6)

**Standard:** 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. (CCSS.Math.Content.HSG-CO.A.7)

**Standard:** Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (CCSS.Math.Content.HSG-CO.A.8)

## About This Chapter

An understanding of congruence will enable students to experiment with transformations in the plane, depicting geometric figures after translations, reflections, or rotations. Students who understand congruence in terms of rigid motions will be able to predict the effect of rigid motions on geometric figures and to describe triangle congruence.

This standard's lessons explain these topics:

- Precise definitions of circle, angle, perpendicular line, parallel line and line segment, based on the undefined notions of point, line, distance along a line and distance around a circular arc.
- Representation of transformations in the plane using transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Comparison of transformations that preserve distance and angle to those that do not (translation versus horizontal stretch).
- Consideration of rectangles, parallelograms, trapezoids or regular polygons and description of the rotations and reflections that carry it onto itself.
- Definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.
- Depiction of a geometric figure and its rotation, reflection or translation, using graph paper, tracing paper or geometry software. Specify a sequence of transformations that will carry a given figure into another.
- Use of 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.
- Explanation of how the criteria of triangle congruence (ASA, SAS and SSS) follow from the definition of congruence in terms of rigid motions.

Students can demonstrate mastery of these subjects by defining basic terms, such as circle, angle and parallel and perpendicular lines. They will be able to represent transformations in the plane using transparencies and geometry software and describe transformations as functions that use some points as inputs and result in others as outputs. The standards and these lessons can prepare them for college classes and careers in fields including architecture, engineering and mathematics.

### How to Use These Lessons in Your Classroom

Here are a few ways you can incorporate these lessons on congruence into your curriculum to help meet the common core standards:

#### Drawing Figures

As a class, watch the lessons on *Drawing Transformed Figures* and *Identifying Sequences of Transformations that Make One Figure into Another*. Assign each student a geometric figure and a translation, rotation or reflection and ask them to depict the figure and show the sequence of the transformation. Share examples with the class.

#### ASA, SAS and SSS Congruence

Have the students view the lesson on * ASA, SAS and SSS Congruence* either as a homework assignment or in class. Ask them to draw triangles that illustrate each of these criteria for congruence.

#### Pre-quiz and post-quiz lessons

Ask the students to take the quiz on *Basic Geometry Vocabulary and Definitions*. Watch the lesson together and review the questions and answers. Have the student re-take the quiz to assess their knowledge.

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### Other Chapters

Other chapters within the Common Core Math - Geometry: High School Standards course