1: Congruence, Proof, and Constructions

1.1: Experiment with transformations in the plane.

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.

Circles
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Inscribed Angles
Parallel, Intersecting, and Skew Lines

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).

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

1.G.CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures

1.G.CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Circles
Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

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.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

1.2: Understand congruence in terms of rigid motions.

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.

Absolute Value with Linear Functions
Circles
Dilations
Holiday Snowflake Designer
Proving Triangles Congruent
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

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.

Proving Triangles Congruent

1.3: Prove geometric theorems.

1.G.CO.9: Prove theorems about lines and angles.

Investigating Angle Theorems

1.G.CO.10: Prove theorems about triangles.

Isosceles and Equilateral Triangles
Proving Triangles Congruent
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Triangle Angle Sum
Triangle Inequalities

1.G.CO.11: Prove theorems about parallelograms.

Parallelogram Conditions
Special Parallelograms

1.4: Make geometric constructions.

1.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.).

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Segment and Angle Bisectors

1.G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

2: Similarity, Proof and Trigonometry

2.1: Understand similarity in terms of similarity transformations.

2.G.SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor:

2.G.SRT.1.a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

Dilations

2.G.SRT.1.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Dilations
Similar Figures

2.G.SRT.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Circles
Dilations
Similar Figures
Similarity in Right Triangles

2.G.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Similar Figures

2.2: Prove theorems involving similarity.

2.G.SRT.4: Prove theorems about triangles.

Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures

2.G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Chords and Arcs
Congruence in Right Triangles
Constructing Congruent Segments and Angles
Dilations
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures
Similarity in Right Triangles

2.3: Define trigonometric ratios and solve problems involving right triangles.

2.G.SRT.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Sine, Cosine, and Tangent Ratios

2.G.SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Cosine Function
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

3: Extending to Three Dimensions

3.1: Explain volume formulas and use them to solve problems.

3.G.GMD.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

3.G.GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Prisms and Cylinders
Pyramids and Cones

4: Connecting Algebra and Geometry Through Coordinates

4.1: Use coordinates to prove simple geometric theorems algebraically.

4.G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Distance Formula

5: Circles With and Without Coordinates

5.1: Understand and apply theorems about circles.

5.G.C.2: Identify and describe relationships among inscribed angles, radii, and chords.

Chords and Arcs
Circumference and Area of Circles
Inscribed Angles

5.G.C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Concurrent Lines, Medians, and Altitudes
Inscribed Angles

5.2: Find arc lengths and areas of sectors of circles.

5.G.C.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Chords and Arcs

5.3: Translate between the geometric description and the equation for a conic section.

5.G.GPE.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

5.5: Apply geometric concepts in modeling situations.

5.G.MG.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Circles