1: Circles

1.2: Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems.

Chords and Arcs
Inscribed Angles

1.3: Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle.

Inscribed Angles

2: Congruence

2.1: Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects.

Circles
Inscribed Angles
Parallel, Intersecting, and Skew Lines

2.2: Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions.

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

2.3: Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations.

Constructing Congruent Segments and Angles

2.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Circles
Rotations, Reflections, and Translations
Similar Figures
Translations

2.6: Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other.

Reflections
Rotations, Reflections, and Translations
Translations

2.7: Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions.

Congruence in Right Triangles
Proving Triangles Congruent

2.8: Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following:

2.8.1: vertical angles are congruent;

Triangle Angle Sum

2.8.2: when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary;

Triangle Angle Sum

2.9: Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following:

2.9.1: measures of interior angles of a triangle sum to 180°;

Isosceles and Equilateral Triangles
Polygon Angle Sum
Triangle Angle Sum

2.9.4: the medians of a triangle meet at a point.

Concurrent Lines, Medians, and Altitudes

2.10: Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following:

2.10.1: opposite sides of a parallelogram are congruent;

Classifying Quadrilaterals
Parallelogram Conditions
Special Parallelograms

2.10.2: opposite angles of a parallelogram are congruent;

Classifying Quadrilaterals
Parallelogram Conditions
Special Parallelograms

2.10.3: diagonals of a parallelogram bisect each other;

Parallelogram Conditions
Special Parallelograms

2.10.4: rectangles are parallelograms with congruent diagonals;

Special Parallelograms

2.11: Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships.

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

3: Geometric Measurement and Dimension

3.1: Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

3.2: Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle.

Prisms and Cylinders
Pyramids and Cones

3.3: Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications.

Prisms and Cylinders
Surface and Lateral Areas of Prisms and Cylinders
Surface and Lateral Areas of Pyramids and Cones

4: Expressing Geometric Properties with Equations

4.1: Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula.

Circles

4.3: Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope.

Cat and Mouse (Modeling with Linear Systems)
Slope
Slope-Intercept Form of a Line

4.5: Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates.

Circles
Distance Formula

6: Similarity, Right Triangles, and Trigonometry

6.1: Understand 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. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Dilations
Similar Figures

6.2: Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other.

Dilations
Perimeters and Areas of Similar Figures
Similar Figures
Similarity in Right Triangles

6.3: Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results.

Similar Figures

6.4: Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following:

6.4.1: A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion.

Similar Figures

6.4.2: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Similar Figures

6.4.3: The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides.

Circles
Cosine Function
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine Function
Surface and Lateral Areas of Pyramids and Cones
Tangent Function

6.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

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

6.6: Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle.

Cosine Function
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

6.8: Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem.

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

7: Interpreting Data

7.1: Select and create an appropriate display, including dot plots, histograms, and box plots, for data that includes only real numbers.

Box-and-Whisker Plots
Correlation
Histograms
Mean, Median, and Mode
Reaction Time 1 (Graphs and Statistics)
Stem-and-Leaf Plots

7.2: Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers.

Box-and-Whisker Plots
Describing Data Using Statistics
Mean, Median, and Mode
Populations and Samples
Reaction Time 1 (Graphs and Statistics)
Real-Time Histogram

7.3: Summarize and represent data from a single data set. Interpret differences in shape, center, and spread in the context of the data set, accounting for possible effects of extreme data points (outliers).

Box-and-Whisker Plots
Describing Data Using Statistics
Least-Squares Best Fit Lines
Mean, Median, and Mode
Populations and Samples
Reaction Time 1 (Graphs and Statistics)
Real-Time Histogram
Stem-and-Leaf Plots