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# 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
14 topics
2.2. Congruence
 2.2.1. Rigid Motions and Congruence 2.2.2. The Angle-Side-Angle (ASA) Criterion for Congruent Triangles 2.2.3. The Side-Angle-Side (SAS) Criterion for Congruent Triangles 2.2.4. The Side-Side-Side (SSS) Criterion for Congruent Triangles
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. Geometric Transformations and Similarity 2.3.6. The Angle-Angle (AA) Criterion for Similar Triangles 2.3.7. The Side-Side-Side (SSS) Criterion for Similar Triangles 2.3.8. The Side-Angle-Side (SAS) Criterion for Similar Triangles 2.3.9. The Midpoint Theorem 2.3.10. The Triangle Proportionality Theorem
3.
Circles
20 topics
3.4. Circles
 3.4.1. Introduction to 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 Using Angles in Degrees 3.4.7. Calculating Areas of Circular Sectors Using Angles in Degrees 3.4.8. Further Calculating Areas of Sectors Using Angles in Degrees 3.4.9. Problem Solving With Circles
 3.5.1. The Inscribed Angle Theorem 3.5.2. Problem Solving Using the Inscribed Angle Theorem 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
 3.6.1. Introduction to Radians 3.6.2. Calculating Arc Length Using Angles in Radians 3.6.3. Calculating Areas of Sectors Using Angles in Radians 3.6.4. Trigonometric Ratios With Radians
4.
Coordinate Geometry
10 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 of Shapes in the Coordinate Plane 4.7.9. Calculating Areas of Rectangles in the Coordinate Plane 4.7.10. Calculating Areas of Triangles and Quadrilaterals in the Coordinate Plane
5.
Triangles & Trigonometry
19 topics
5.8. Right Triangles
 5.8.1. The Pythagorean Theorem 5.8.2. The 45-45-90 Triangle 5.8.3. The 30-60-90 Triangle 5.8.4. The Area of a 45-45-90 Triangle 5.8.5. The Area of a 30-60-90 Triangle 5.8.6. The Area of an Equilateral Triangle 5.8.7. Diagonals of Squares
5.9. Introduction to Trigonometry
 5.9.1. Angles and Sides in Right Triangles 5.9.2. The Trigonometric Ratios 5.9.3. Calculating Trigonometric Ratios Using the Pythagorean Theorem 5.9.4. Calculating Side Lengths of Right Triangles Using Trigonometry 5.9.5. Calculating Angles in Right Triangles Using Trigonometry 5.9.6. Modeling With Trigonometry 5.9.7. The Reciprocal Trigonometric Ratios 5.9.8. Trigonometric Ratios in Similar Right Triangles 5.9.9. Trigonometric Functions of Complementary Angles 5.9.10. Special Trigonometric Ratios 5.9.11. Calculating the Area of a Right Triangle Using Trigonometry 5.9.12. Solving Multiple Right Triangles Using Trigonometry
6.
Solid Geometry
25 topics
6.10. Introduction to Solid Geometry
 6.10.1. Identifying Three-Dimensional Shapes 6.10.2. Faces, Vertices, and Edges of Polyhedrons 6.10.3. Nets of Polyhedrons 6.10.4. Finding Surface Areas Using Nets 6.10.5. The Distance Formula in Three Dimensions 6.10.6. Euler's Formula for Polyhedra 6.10.7. The Five Platonic Solids
6.11. Rectangular Solids and Pyramids
 6.11.1. Volumes of Cubes 6.11.2. Surface Areas of Cubes 6.11.3. Face Diagonals of Cubes 6.11.4. Diagonals of Cubes 6.11.5. Volumes of Rectangular Solids 6.11.6. Surface Areas of Rectangular Solids 6.11.7. Diagonals of Rectangular Solids 6.11.8. Volumes of Pyramids 6.11.9. Surface Areas of Pyramids
6.12. Non-Polyhedrons
 6.12.1. Volumes of Cylinders 6.12.2. Surface Areas of Cylinders 6.12.3. Volumes of Right Cones 6.12.4. Slant Heights of Right Cones 6.12.5. Surface Areas of Right Cones 6.12.6. Volumes of Spheres 6.12.7. Surface Areas of Spheres 6.12.8. Conical Frustums 6.12.9. Volumes of Revolution