(1) 
GCO.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 15

11 Points, Lines and Planes
12 Linear Measure
13 Distance and Midpoints
14 Angle Measure


*1a denotes Mid  Module Assessment
*[1a] denotes End  of Module Assessment
1,3a,3b

(2) 
GCO.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 15

12 Linear Measure
13 Distance and Midpoints
14 Angle Measure
15 Angle Relationships
1  6 Extend: Geometry Software Lab – TwoDimensional Figures


3a, 3b, 4

(2) 
GCO.13 
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. 

(8) 
8.G.2 
Understand that a twodimensional 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 15

105 Extend: Geometry Lab Inscribed Circles (omit Circumscribed Circles)


[5a,5b]







(2) 
GCO.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 angleangle 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 611

27 Proving Segment Relationships
28 Proving Angle Relationships
ADDITIONAL CONTENT: Isosceles Triangle Theorem/Converse


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







(1) 
GCO.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 twodimensional 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 twodimensional figures using coordinates. 

Lessons 1221

94 Explore: Geometry Software LabCompositions of Transformations
94 Compostions of Transformations
96 Dilations

Transformation, Isometry

2a, 2b, 2c
[1a,1b]

(1) 
GCO.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 twodimensional 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 1221

95 Symmetry

Reflection, Rotation

[5a,5b]

(1) 
GCO.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 twodimensional 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 1221

91 Reflections
92 Translations
93 Rotations

Translation, Reflection, Rotation

5a, 5b, 5c

(1) 
GCO.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 1221

91 Reflections
92 Translations
93 Explore: Geometry LabRotations
93 Rotations
94 Explore: Geometry Software LabCompositions of Transformations
94 Compositions of Transformations

Translation, Reflection, Rotation

4

(1) 
GCO.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 twodimensional 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 1221

47 Explore: Graphing Technology LabCongruence Transformations
47 Congruence Transformations

Congruence

6a, 6b, 6c

(2) 
GCO.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 twodimensional 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 1221

47 Congruence Transformations

Congruence

[4a,4b]

(2) 
GCO.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 1221

51 Explore: Geometry LabConstructing Bisectors
52 Explore: Geometry LabConstructing Medians and Altitudes
55 Explore: Graphing Tech. LabTriangle Inequality


3a, 3b, 4







(2) 
GCO.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 twodimensional 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 2227

43 Congruent Triangles
47 Congruence Transformations

Congruence

[4a,4b]

(1) 
GCO.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 twodimensional 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 2227

96 Extend: Geometry LabEstablishing Triangle Congruence and Similarity

Congruence

[4a,4b]







(2) 
GCO.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 angleangle 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 2830

32 Angles and Parallel Lines
35 Proving Parallel Lines


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

(1) 
GCO.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 angleangle 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 2830

42 Angles of Triangles
44 Proving Triangles CongruentSSS, SAS
45 Proving Triangles CongruentASA, AAS
ADDITIONAL CONTENT: HL (HypotenuseLeg)
46 Isosceles and Equilateral Triangles
ADDITIONAL CONTENT: Be sure to include mention of Isosceles Triangle Theorem/Converse
48 Triangles and Coordinate Proof


[2,3]

(1) 
GCO.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 angleangle 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 2830

62 Parallelograms
63 Tests for Parallelograms
64 Rectangles
65 Rhombi and Squares


[6]







(2) 
GCO.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 3132

105 Extend: Geometry LabInscribed and Circumscribed Circles


[5a,5b]



Lessons 3334

Review of the Assumptions


End of Module Assessment (lessons 134)
