Professor Noggle's lectures on geometry are exceptionally clear and well organized. He has an evident love for the topic, and a real gift for conveying the elegance and precision of geometric concepts and demonstrations. You will learn how geometrical concepts link new theorems and ideas to previous ones. This helps you see geometry as a unified body of knowledge whose concepts build upon one another.
High School Level—Geometry
Develop the ability to read, write, think, and communicate about the concepts of high school level geometry.
Overview Course No. 105
About
Professor James Noggle is a math instructor at Pendleton Heights High School in Pendleton, Indiana, where he has been teaching for more than 30 years. The math courses in which he has specialized are algebra I, geometry, trigonometry, and analytical geometry. This range of courses has enabled him to help his students see a broader view of their math and relate it in many ways to the uses in applications at more advanced levels. Professor Noggle graduated from Anderson University in Anderson, Indiana, with a math major, and he earned his M.A.E. with a math major from Ball State University. He is the recipient of Tandy Technology Scholar Award for academic excellence in mathematics, science, and computer science.
01: Fundamental Geometric Concepts
In this introductory lesson, we define point, line, and plane; use and understand the terms space, collinear, intersection, segment, and ray; learn terminology of various expressions relative to points, lines, and planes; and establish a system of linear measurement.
33 min
02: Angles and Angle Measure
We explore the definition of an angle and learn its parts; establish a system of angle measurement; recognize and classify types of angles; and show angle relationships.
31 min
03: Inductive Reasoning and Deductive Reasoning
We use inductive reasoning to discover mathematical relationships, recognize real-world applications of inductive reasoning, and understand conditional statements and deductive reasoning.
32 min
04: Preparing Logical Reasons for a Two-Column Proof
We review properties of equality for real numbers; summarize and review postulates related to points, lines, planes, and angles; and introduce new theorems related to points, lines, planes, and angles.
31 min
05: Planning Proofs in Geometry
We discuss the key elements of a two-column proof; learn how to draw and label a diagram for a proof; write a plan for the proof; use strategy to write a two-column proof; and write a two-column proof.
32 min
06: Parallel Lines and Planes
We identify parallel lines, skew lines, parallel planes, transversals, and the angles formed by them; and we state and apply postulates and theorems about angles formed when parallel lines are intersected by a transversal.
31 min
07: Triangles
We classify triangles according to their sides and angles, and use theorems about the angles of a triangle.
32 min
08: Polygons and Their Angles
We distinguish between convex polygons and concave polygons; name convex and regular polygons; and find measures of interior and exterior angles of convex polygons.
31 min
09: Congruence of Triangles
We identify congruent parts of congruent triangles; state and apply the SSS, SAS, and ASA postulates; and use those postulates to prove triangles congruent.
31 min
10: Variations of Congruent Triangles
We deduce that segments or angles are congruent by first proving two triangles congruent; use two congruent triangles to prove other, related facts; and prove two triangles congruent by first proving two other triangles congruent.
30 min
11: More Theorems Related to Congruent Triangles
We use the isosceles triangle theorem, its converse, and related theorems; and use the AAS theorem and right triangle theorems to prove triangles congruent.
31 min
12: Median, Altitudes, Perpendicular Bisectors, and Angle Bisectors
We discuss definitions of median, altitude, perpendicular bisector, angle bisector, and related terms; state and apply theorems related to them; and learn their points of concurrence.
31 min
13: Parallelograms
We state and apply the definition of a parallelogram, state and apply theorems related to the properties of a parallelogram, and prove that certain quadrilaterals are parallelograms.
31 min
14: Rectangles, Rhombuses, and Squares
We identify rectangles, rhombuses, and squares; and state and apply properties and theorems related to their properties.
31 min
15: Trapezoids, Isosceles Trapezoids, and Kites
We learn to identify trapezoids, isosceles trapezoids, and kites, and we state and apply properties and theorems related to them.
31 min
16: Inequalities in Geometry
We review properties of inequality for real numbers and relate them to segments and angles; state and apply the inequality relations for one triangle and for two triangles.
31 min
17: Ratio, Proportion, and Similarity
In this lesson we explain how to express a ratio in its simplest form; identify, write, and solve proportions; use ratios and proportions to solve problems; express a given proportion in other equivalent forms; and apply the properties of similar polygons using ratios and proportions.
31 min
18: Similar Triangles
We state and apply the AA Similarity Postulate, the SAS Similarity Theorem, and the SSS Similarity Theorem. We learn to solve for unknown measurements using the new postulates and theorems related to similarity and to apply the Triangle Proportionality Theorem, the Triangle Angle-Bisector Theorem, and related theorems.
32 min
19: Right Triangles and the Pythagorean Theorem
We apply proportions and the concepts of proportionality in right triangles, use and apply the geometric mean between two values, state and apply the relationships that exist when the altitude of a triangle is drawn to the hypotenuse, state and apply the Pythagorean Theorem and its converse, and relate the Pythagorean Theorem to inequalities.
31 min
20: Special Right Triangles
We explore how to apply relationships in a 45°-45°-90° right triangle and in a 30°-60°-90° right triangle and use those relationships in the development of the unit circle.
30 min
21: Right-Triangle Trigonometry
We define and apply the tangent, sine, and cosine ratios for an acute angle and solve right-triangle problems using those ratios.
31 min
22: Applications of Trigonometry in Geometry
We address how to select the correct trigonometric ratio to use in problem solving, and how to use trigonometry to solve real-life problems.
32 min
23: Tangents, Arcs, and Chords of a Circle
We apply basic definitions and concepts related to circles, and state and apply properties and theorems regarding circles and their tangents, chords, central angles, and arcs.
31 min
24: Angles and Segments of a Circle
We apply basic definitions and theorems related to inscribed angles; state and apply theorems involving angles with vertices not on the circle formed by tangents, chords, and secants; and state and apply theorems involving lengths of chords, secant segments, and tangent segments.
30 min
25: The Circle as a Whole and Its Parts
We state and apply the formulas for the circumference and area of a circle, and for the arc lengths and the areas of sectors of a circle.
31 min
26: The Logic of Constructions through Applied Theorems (Part I)
In sample exercises, we review lessons and solve problems having to do with segments, angles, parallel and perpendicular lines, circles and arcs, and others.
30 min
27: The Logic of Constructions through Applied Theorems (Part II)
Continuing sample exercises, we review lessons and solve problems having to do with triangles, isosceles triangles, proportions, hexagons, and others.
31 min
28: Areas of Polygons
We address the derivation of the area formulas and apply those formulas to find the areas of a rectangle, square, parallelogram, triangle, trapezoid, and regular polygon.
31 min
29: Prisms, Pyramids, and Polyhedra
We explore definitions of a polyhedron, prism, pyramid, and related terms; understand the logical derivation of area and volume formulas; and apply theorems to compute the lateral area, total area, and volume of prisms and pyramids.
32 min
30: Cylinders, Cones, and Spheres
We explain the definitions of cylinder, cone, and sphere; explain the logical derivation of area and volume formulas; and apply theorems to compute the lateral areas, total areas, and volumes of cylinders, cones, and spheres.
31 min
