High School Level - Geometry [repost]

For more than 30 years, Professor James Noggle has been letting his students in on the secret to making the mysteries of lines, planes, angles, inductive and deductive reasoning, parallel lines and planes, triangles, polygons, and other geometric concepts easy to grasp. And in his course, Geometry, you'll develop the ability to read, write, think, and communicate about the concepts of geometry. As your comprehension and understanding of the geometrical vocabulary increase, you will have the ability to explain answers, justify mathematical reasoning, and describe problem-solving strategies.

Professor Noggle relies heavily on the blackboard and a flipchart on an easel in his 30 lectures. Very little use is made of computer-generated graphics, though several physical models of geometric objects are used throughout the lectures.

A New Way to Look at the World around You

The language of geometry is beautifully expressed in words, symbols, formulas, postulates, and theorems. These are the dynamic tools by which you can solve problems, communicate, and express geometrical ideas and concepts.

Connecting the geometrical concepts includes linking new theorems and ideas to previous ones. This helps you to see geometry as a unified body of knowledge whose concepts build upon one another. And you should be able to connect these concepts to appropriate real-world applications.

Professor Noggle’s Geometry course will begin with basic fundamental concepts used throughout the course. Students will be able to recognize and define such terms as points, planes, and angles; parallel lines, skew lines, parallel planes, and transversals; as well as the terms space, collinear, intersection, segment, and ray.

Students will discover the world of angles—symbols used for them, establishing a system of angle measurement, classifying the different types, and showing angle relationships.

The course then continues with the use of inductive reasoning to discover mathematical relationships and recognize real-world applications of inductive reasoning, conditional statements, and deductive reasoning.

Using the Fundamental Tools of Geometry

After the first few lectures introduce students to the basic terms, Professor Noggle will open the world of geometry to students. Upon completion of this course, you should be able to:

Classify triangles according to their sides and angles
Distinguish between convex polygons and concave polygons, and find the interior and exterior angles of convex polygons
State and apply postulates and theorems involving parallel lines and convex polygons to solve related problems and prove statements using deductive reasoning.
Explain the ratio in its simplest form; identify, write, and solve proportions
Identify congruent parts of congruent triangles; state and apply the SSS, SAS, and ASA postulates; and use those postulates to prove triangles congruent
Be able to define, state, and apply theorems for parallelograms, rectangles, rhombuses, squares, and trapezoids
Apply proportions and concepts of proportionality in right triangles; discuss the Pythagorean Theorem
Explore the relationships between right and isosceles triangles
Define tangent, sine, and cosine rations for angles
State and apply properties and theorems regarding circles and their tangents, chords, central angles, and arcs
Address the derivation of the area formulas and apply those formulas to rectangles, squares, parallelograms, triangles, trapezoids, and regular polygons
Define polyhedron, prism, pyramid, cylinder, cone, and sphere; and apply theorems to compute the lateral area, total area, and volume of the prism, pyramid, cylinders, cones, and spheres.

Lectures:

1. Fundamental Geometric Concepts
2. Angles and Angle Measure
3. Inductive Reasoning and Deductive Reasoning
4. Preparing Logical Reasons for a Two-Column Proof
5. Planning Proofs in Geometry
6. Parallel Lines and Planes
7. Triangles
8. Polygons and Their Angles
9. Congruence of Triangles
10. Variations of Congruent Triangles
11. More Theorems Related to Congruent Triangles
12. Median, Altitudes, Perpendicular Bisectors, and Angle Bisectors
13. Parallelograms
14. Rectangles, Rhombuses, and Squares
15. Trapezoids, Isosceles Trapezoids, and Kites
16. Inequalities in Geometry
17. Ratio, Proportion, and Similarity
18. Similar Triangles
19. Right Triangles and the Pythagorean Theorem
20. Special Right Triangles
21. Right-Triangle Trigonometry
22. Applications of Trigonometry in Geometry
23. Tangents, Arcs, and Chords of a Circle
24. Angles and Segments of a Circle
25. The Circle as a Whole and Its Parts
26. The Logic of Constructions through Applied Theorems (Part I)
27. The Logic of Constructions through Applied Theorems (Part II)
28. Areas of Polygons
29. Prisms, Pyramids, and Polyhedra
30. Cylinders, Cones, and Spheres

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