# Geometry

This fully accredited, Common Core-aligned Geometry course builds on existing knowledge gained in Algebra I to provide students with a comprehensive introduction to high-school geometry. This course lays the foundations for progression onto Algebra II, where more advanced geometrical and trigonometric concepts are encountered.

## Content

In this comprehensive course, students will delve deeply into the world of geometric transformations in the plane. They will construct functions that represent specific transformations and apply these functions to manipulate various geometric objects. Moreover, students will build upon their knowledge of reflective and rotational symmetry, further enriching their understanding of these fundamental concepts.

The course introduces students to the concept of rigid motion as a transformation that preserves both distances and angles. Students will learn to identify whether a given transformation qualifies as a rigid motion and will describe the concept of congruence in terms of these rigid motions.

Progressing from the concept of congruence, students will master the notion of similarity. They will learn to formally describe similarity through the use of similarity transformations and will apply these concepts to solve problems involving similar polygons.

In addition, students will become proficient in using triangle congruence and similarity criteria, and they will apply key theorems, such as the midpoint and triangle proportionality theorems, deepening their comprehension of these critical geometric principles.

Within this course, a particular emphasis is placed on enhancing studentsâ€™ understanding of circles. Students will learn to articulate the precise properties of circles and calculate the areas and circumferences of circles and circular sectors described using both degrees and radians. Advanced circle topics are also covered, including the Inscribed Angle Theorem, Thalesâ€™ Theorem, and properties of tangents to circles. Moreover, students will tackle multifaceted problems, including those set within real-world contexts, synthesizing multiple concepts they have learned.

Much of this course is dedicated to extending students' geometric knowledge to problems within the coordinate plane. Students will derive equations of parallel and perpendicular lines, calculate distances and midpoints in the plane, and apply this knowledge to compute areas and perimeters of polygons.

Deepening students' understanding of right triangles is a key objective of this course, setting a foundation vital for higher-level mathematics. Students will become proficient in applying the Pythagorean Theorem and understanding its role in deducing properties of special right triangles. This course marks students' initial encounter with trigonometric ratios, equipping them to solve problems that require computing angles and side lengths of right triangles and applying these skills in more complex scenarios.

In the concluding unit, students will expand their geometric knowledge beyond the plane, exploring both polyhedra and non-polyhedra. They will learn to calculate volumes and surface areas and will delve into the properties of spheres, cylinders, cones, and pyramids. Advanced concepts, such as Eulerâ€™s formula for polyhedra, Platonic solids, and volumes of revolution, will also be introduced, adding a sophisticated layer to students' understanding of geometry.

Upon successful completion of this course, students will have mastered the following:
• Perform translations, rotations, reflections, dilations, and stretches of geometric figures, including coordinate plane transformations, and describe them using functions.
• Find images of geometric objects under the action of composite transformations.
• Identify and analyze reflective and rotational symmetry in geometric figures.
• Understand the concept of a rigid motion, and use rigid motions to describe congruence between geometric figures.
• Apply the ASA, SAS, and SSS congruence criteria and the AA, SSS, and SAS similarity criteria.
• Solve problems involving similar polygons, including working with area scale factors, and the midpoint and triangle proportionality theorems.
• Form precise definitions of circles, interiors and exteriors of circles, circular arcs, sectors, and segments.
• Calculate the circumference and area of circles and circular sectors, and solve related problems.
• Understand and apply the Inscribed Angle Theorem, Thales' Theorem, work with inscribed quadrilaterals, and understand the properties of tangent lines to circles.
• Find equations of parallel and perpendicular lines.
• Apply the distance and midpoint formulas.
• Calculate perimeters and areas of polygons in the coordinate plane.
• Understand and apply the Pythagorean Theorem and its converse.
• Derive and apply properties of 45-45-90 and 30-60-90 triangles.
• Calculate side lengths and angle measures in right triangles using trigonometric ratios.
• Use trigonometry to model and solve real-world and mathematical problems, including calculating areas in right triangles.
• Understand that radians provide an alternative way of measuring angles that's often more convenient, convert between radians and degrees, and calculate arc lengths, areas of sectors, and trigonometric ratios using radians.
• Understand and apply properties of complementary angles in right triangles.
• Calculate the surface area and volume for various three-dimensional figures, including cubes, rectangular solids, pyramids, cylinders, cones, and spheres.
• Apply Euler's Formula for Polyhedra and recognize the characteristics of the five Platonic Solids.
• Apply the distance formula in three dimensions.
• Understand and calculate volumes of revolution.
1.
Geometric Transformations
12 topics
1.1. Geometric Transformations
 1.1.1. Translations of Geometric Figures 1.1.2. Rotations of Geometric Figures 1.1.3. Rotating Objects in the Coordinate Plane Using Functions 1.1.4. Reflections of Geometric Figures in the Cartesian Plane 1.1.5. Reflections of Figures Across Arbitrary Lines 1.1.6. Dilations of Geometric Figures 1.1.7. Dilations of Figures in the Coordinate Plane 1.1.8. Stretches of Geometric Figures 1.1.9. Combining Stretches of Geometric Figures 1.1.10. Combining Geometric Transformations 1.1.11. Reflective Symmetry 1.1.12. Rotational Symmetry
2.
Congruence & Similarity
16 topics
2.2. Congruence
 2.2.1. Rigid Motions and Congruence 2.2.2. The ASA Congruence Criterion 2.2.3. The AAS Congruence Criterion 2.2.4. The SAS Congruence Criterion 2.2.5. The SSS Congruence Criterion 2.2.6. The HL Congruence Criterion
2.3. Similarity
 2.3.1. Similarity and Similar Polygons 2.3.2. Side Lengths and Angle Measures of Similar Polygons 2.3.3. Areas of Similar Polygons 2.3.4. Working With Areas of Similar Polygons 2.3.5. Similarity Transformations 2.3.6. The AA Similarity Criterion 2.3.7. The SSS Similarity Criterion 2.3.8. The SAS Similarity Criterion 2.3.9. The Midpoint Theorem 2.3.10. The Triangle Proportionality Theorem
3.
20 topics
3.4. Circles
 3.4.1. Circles 3.4.2. Arcs, Segments, and Sectors of Circles 3.4.3. The Circumference of a Circle 3.4.4. Areas of Circles 3.4.5. Central Angles and Arcs 3.4.6. Calculating Arc Lengths of Circular Sectors 3.4.7. Calculating Areas of Circular Sectors 3.4.8. Further Calculating Areas of Sectors 3.4.9. Problem Solving With Circles
3.5. Circle Theorems
 3.5.1. Inscribed Angles 3.5.2. Problem-Solving With Inscribed Angles 3.5.3. Thales' Theorem 3.5.4. Angles in Inscribed Right Triangles 3.5.5. Inscribed Quadrilaterals 3.5.6. Tangent Lines to Circles 3.5.7. Circle Similarity
4.
Geometry in the Coordinate Plane
16 topics
4.7. Coordinate Geometry
 4.7.1. Parallel Lines in the Coordinate Plane 4.7.2. Finding the Equation of a Parallel Line 4.7.3. Perpendicular Lines in the Coordinate Plane 4.7.4. Finding Equations of Perpendicular Lines 4.7.5. Midpoints in the Coordinate Plane 4.7.6. The Distance Formula 4.7.7. The Shortest Distance Between a Point and a Line 4.7.8. Calculating Perimeters in the Plane 4.7.9. Calculating Areas of Rectangles in the Plane 4.7.10. Calculating Areas of Triangles and Quadrilaterals in the Plane
4.8. Circles in the Coordinate Plane
 4.8.1. Circles in the Coordinate Plane 4.8.2. Equations of Circles Centered at the Origin 4.8.3. Equations of Circles 4.8.4. Determining Properties of Circles by Completing the Square 4.8.5. Calculating Circle Intercepts 4.8.6. Intersections of Circles with Lines
5.
Triangles & Trigonometry
19 topics
5.9. Right Triangles
 5.9.1. The Pythagorean Theorem 5.9.2. The 45-45-90 Triangle 5.9.3. The 30-60-90 Triangle 5.9.4. The Area of a 45-45-90 Triangle 5.9.5. The Area of a 30-60-90 Triangle 5.9.6. The Area of an Equilateral Triangle 5.9.7. Diagonals of Squares
5.10. Trigonometry
 5.10.1. Angles and Sides in Right Triangles 5.10.2. The Trigonometric Ratios 5.10.3. Calculating Trigonometric Ratios Using the Pythagorean Theorem 5.10.4. Calculating Side Lengths of Right Triangles Using Trigonometry 5.10.5. Calculating Angles in Right Triangles Using Trigonometry 5.10.6. Modeling With Trigonometry 5.10.7. The Reciprocal Trigonometric Ratios 5.10.8. Trigonometric Ratios in Similar Right Triangles 5.10.9. Trigonometric Functions of Complementary Angles 5.10.10. Special Trigonometric Ratios 5.10.11. Calculating the Area of a Right Triangle Using Trigonometry 5.10.12. Solving Multiple Right Triangles Using Trigonometry
6.
Solid Geometry
25 topics
6.11. Introduction to Solid Geometry
 6.11.1. Identifying Three-Dimensional Shapes 6.11.2. Faces, Vertices, and Edges of Polyhedrons 6.11.3. Nets of Polyhedrons 6.11.4. Finding Surface Areas Using Nets 6.11.5. The Distance Formula in Three Dimensions 6.11.6. Euler's Formula for Polyhedra 6.11.7. The Five Platonic Solids
6.12. Rectangular Solids and Pyramids
 6.12.1. Volumes of Cubes 6.12.2. Surface Areas of Cubes 6.12.3. Face Diagonals of Cubes 6.12.4. Diagonals of Cubes 6.12.5. Volumes of Rectangular Solids 6.12.6. Surface Areas of Rectangular Solids 6.12.7. Diagonals of Rectangular Solids 6.12.8. Volumes of Pyramids 6.12.9. Surface Areas of Pyramids
6.13. Non-Polyhedrons
 6.13.1. Volumes of Cylinders 6.13.2. Surface Areas of Cylinders 6.13.3. Volumes of Right Cones 6.13.4. Slant Heights of Right Cones 6.13.5. Surface Areas of Right Cones 6.13.6. Volumes of Spheres 6.13.7. Surface Areas of Spheres 6.13.8. Conical Frustums 6.13.9. Volumes of Revolution
7.
Probability & Combinatorics
31 topics
7.14. Introduction to Probability
 7.14.1. Sets 7.14.2. Probability From Experimental Data 7.14.3. Sample Spaces and Events in Probability 7.14.4. Single Events in Probability 7.14.5. The Complement of an Event 7.14.6. Venn Diagrams in Probability 7.14.7. Geometric Probability
7.15. Compound Events in Probability
 7.15.1. The Union of Sets 7.15.2. The Intersection of Sets 7.15.3. Compound Events in Probability From Experimental Data 7.15.4. Computing Probabilities for Compound Events Using Venn Diagrams 7.15.5. Computing Probabilities of Events Containing Complements Using Venn Diagrams 7.15.6. Computing Probabilities for Three Events Using Venn Diagrams 7.15.7. The Addition Law of Probability 7.15.8. Applying the Addition Law With Event Complements 7.15.9. Mutually Exclusive Events
7.16. Conditional Probability
 7.16.1. Conditional Probabilities From Venn Diagrams 7.16.2. Conditional Probabilities From Tables 7.16.3. The Multiplication Law for Conditional Probability 7.16.4. The Law of Total Probability 7.16.5. Tree Diagrams for Dependent Events 7.16.6. Tree Diagrams for Dependent Events: Applications 7.16.7. Independent Events 7.16.8. Tree Diagrams for Independent Events
7.17. Combinatorics
 7.17.1. The Rule of Sum and the Rule of Product 7.17.2. Factorials 7.17.3. Factorials in Variable Expressions 7.17.4. Ordering Objects 7.17.5. Permutations 7.17.6. Combinations 7.17.7. Computing Probabilities Using Combinatorics