Last updated: 7/14/2016

## Mat230 Geometry CCLS - Module 1-Congruence, Proof, and Constructions (45 days - 10 weeks)

Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
 (1) G-CO.1 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.

Lesson 1-5

1-1 Points, Lines and Planes

1-2 Linear Measure

1-3 Distance and Midpoints

1-4 Angle Measure

*1a denotes Mid - Module Assessment

*[1a] denotes End - of Module Assessment

1,3a,3b

 (2) G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Lessons 1-5

1-2 Linear Measure

1-3  Distance and Midpoints

1-4  Angle Measure

1-5  Angle Relationships

1 - 6 Extend: Geometry Software Lab – Two-Dimensional Figures

3a, 3b, 4

 (2) G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
 (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lessons 1-5

10-5 Extend: Geometry Lab Inscribed Circles (omit Circumscribed Circles)

[5a,5b]

 (2) G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and orresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
 (4) 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Lessons 6-11

2-7 Proving Segment Relationships

2-8 Proving Angle Relationships

3a, 3b, 6a, 6b, 6c

 (1) G-CO.2 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).
 (6) 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: (6) 8.G.1.a Lines are taken to lines, and line segments to line segments of the same length. (6) 8.G.1.b Angles are taken to angles of the same measure. (6) 8.G.1.c Parallel lines are taken to parallel lines. (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. (1) 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Lessons 12-21

9-4 Explore: Geometry Software Lab-Compositions of Transformations

9-4 Compostions of Transformations

9-6 Dilations

Transformation, Isometry

2a, 2b, 2c

[1a,1b]

 (1) G-CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
 (6) 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: (6) 8.G.1.a Lines are taken to lines, and line segments to line segments of the same length. (6) 8.G.1.b Angles are taken to angles of the same measure. (6) 8.G.1.c Parallel lines are taken to parallel lines. (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lessons 12-21

9-5 Symmetry

Reflection, Rotation

[5a,5b]

 (1) G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
 (6) 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: (6) 8.G.1.a Lines are taken to lines, and line segments to line segments of the same length. (6) 8.G.1.b Angles are taken to angles of the same measure. (6) 8.G.1.c Parallel lines are taken to parallel lines. (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lessons 12-21

9-1 Reflections

9-2 Translations

9-3 Rotations

Translation, Reflection, Rotation

5a, 5b, 5c

 (1) G-CO.5 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.
 (6) 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: (6) 8.G.1.a Lines are taken to lines, and line segments to line segments of the same length. (6) 8.G.1.b Angles are taken to angles of the same measure. (6) 8.G.1.c Parallel lines are taken to parallel lines.

Lessons 12-21

9-1 Reflections

9-2 Translations

9-3 Explore: Geometry Lab-Rotations

9-3 Rotations

9-4 Explore: Geometry Software Lab-Compositions of Transformations

9-4 Compositions of Transformations

Translation, Reflection, Rotation

4

 (1) G-CO.6 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.
 (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lessons 12-21

4-7 Explore: Graphing Technology Lab-Congruence Transformations

4-7 Congruence Transformations

Congruence

6a, 6b, 6c

 (2) G-CO.7 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.
 (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lessons 12-21

4-7 Congruence Transformations

Congruence

[4a,4b]

 (2) G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
 (6) 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: (6) 8.G.1.a Lines are taken to lines, and line segments to line segments of the same length. (6) 8.G.1.b Angles are taken to angles of the same measure. (6) 8.G.1.c Parallel lines are taken to parallel lines.

Lessons 12-21

5-1 Explore: Geometry Lab-Constructing Bisectors

5-2 Explore: Geometry Lab-Constructing Medians and Altitudes

5-5 Explore: Graphing Tech. Lab-Triangle Inequality

3a, 3b, 4

 (2) G-CO.7 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.
 (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lessons 22-27

4-3 Congruent Triangles

4-7 Congruence Transformations

Congruence

[4a,4b]

 (1) G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
 (8) 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Lessons 22-27

9-6 Extend: Geometry Lab-Establishing Triangle Congruence and Similarity

Congruence

[4a,4b]

 (2) G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and orresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
 (4) 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Lessons 28-30

3-2 Angles and Parallel Lines

3-5 Proving Parallel Lines

3a, 3b, 6a, 6b, 6c

 (1) G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
 (4) 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Lessons 28-30

4-2 Angles of Triangles

4-4 Proving Triangles Congruent-SSS, SAS

4-5 Proving Triangles Congruent-ASA, AAS

4-6 Isosceles and Equilateral Triangles

ADDITIONAL CONTENT: Be sure to include mention of Isosceles Triangle Theorem/Converse

4-8 Triangles and Coordinate Proof

[2,3]

 (1) G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
 (4) 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Lessons 28-30

6-2 Parallelograms

6-3 Tests for Parallelograms

6-4 Rectangles

6-5 Rhombi and Squares

[6]

 (2) G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
 (6) 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: (6) 8.G.1.a Lines are taken to lines, and line segments to line segments of the same length. (6) 8.G.1.b Angles are taken to angles of the same measure. (6) 8.G.1.c Parallel lines are taken to parallel lines.

Lessons 31-32

10-5 Extend: Geometry Lab-Inscribed and Circumscribed Circles

[5a,5b]

Lessons 33-34

Review of the Assumptions

End of Module Assessment (lessons 1-34)