Life of Fred Geometry Expanded


One day in Fred’s life in which he . . . • Falls in Love! • Teaches you how to write an opera! • Buys Hecks Kitchen! • Does all of geometry up to the 14th dimension!

Geometry is one course that is different from all the rest. In the other courses, the emphasis is on calculating, manipulating and computing answers. In contrast, in geometry there are proofs to be created. It is much more like solving puzzles than grinding out numerical answers. For example, if you start out with a triangle that has two sides of equal length, you are asked to show that it has two angles that have the same size.

You will need at least one year of high school algebra. (Life of Fred: Beginning Algebra). In this geometry book for example, on p. 26 we will go from y + y = z to y = ½ z. It is preferable, however, that you have completed the two years of high school algebra. (Life of Fred: Beginning Algebra and Life of Fred: Advanced Algebra Expanded Edition). Most schools stick geometry between the two years of algebra – beginning algebra, geometry, advanced algebra – but there are a couple of reasons why this is not the best approach.

First, when you stick geometry between the two algebra courses, you will have a whole year to forget beginning algebra. Taking advanced algebra right after beginning algebra keeps the algebra fresh.

Second, the heart of geometry is learning how to do proofs. This requires an “older mind” than the mechanical stuff in the algebra courses. A person’s brain develops in stages. Most three-year-olds don’t enjoy quiz shows on television.

In this course you will learn about… Non-Euclidean Geometry Attempts to prove the Parallel Postulate Nicolai Ivanovich Lobachevsky’s geometry Consistent Mathematical theories Georg Friedrich Bernhard Riemann’s geometry Geometries with only three points Points and Lines Attempts to prove the parallel postulate Collinear points Concurrent lines Coplanar lines Coordinates of a point Definition of when one point is between two other points Exterior Angles Indirect Lines Line segments Midpoint Parallel lines Perpendicular Lines Perpendicular Bisectors Postulates and theorems Skew lines Distance from a point to a Line Tangent and secant lines Theorems, propositions, lemmas, and corollaries Undefined terms Quadrilaterals Honors Problem of the century: If two angle bisectors are congruent when drawn to the opposite sides, then the triangle is isosceles Intercepted segment Kite Midsegment of a triangle Parallelogram Rectangle Rhombus Square Trapezoid Solid Geometry Euler’s theorem A line perpendicular to a plane Distance from a point to a plane Parallel and perpendicular planes Polyhedrons hexahedron (cube) tetrahedron Octahedron Icosahedron Dodecahderon Volume Formulas: cylinders, prisms, cones, pyramids, spheres Cavalieri’s Principle Lateral Surface Area Symbolic Logic Contrapositives If…then…statements Truth tables Triangles Acute and Obtuse Triangles Adjacent, opposite, hypotenuse Altitudes Angle bisector theorem Definition of a triangle Drawing auxiliary lines equilateral and equiangular triangles Hypotenuse-leg theorem Isosceles triangle theorem Medians Pons Asinorum Proof that a right angle is congruent to an obtuse angle using euclidean geometry Proportions Right Triangles Scalene Triangles Similar triangles SSS, SAS, ASA postulates Angles Acute, obtuse, and right angles Alternate interior angles and corresponding angles Congruent angles Degrees, minutes, and seconds Euclid’s The Elements Exterior angles Inscribed angle theorem Linear pairs Rays Supplementary angles Two proofs of the exterior angle theorem Vertical angles Area Area and volume formulas Heron’s Formula Parallelograms Perimeter Polygons Pythagorean Theorem Rectangles, Rhombuses, and Squares Trapezoids Triangle inequality Triangles Circles Center, radius, chord, diameter, secant, tangent Concentric circles Central Angles Circumference Arcs Inscribed angles Proof by Cases Sectors Constructions Compass and straightedge Rules of the Game Rusty compass constructions Golden Rectangles and golden ratio Trisecting an angle and squaring a circle Incenter and circumcenter of a triangle Collapsible compass constructions 46 popular constructions Coordinate Geometry Analytic geometry Cartesian/rectangular/orthogonal coordinate system Axes, origins, and quadrants slope distance formula midpoint formula proofs using analytic geometry Flawless (Modern) Geometry Proof that every triangle is isosceles Proof that an obtuse angle is congruent to a right angle 19-year-old Robert L Moore’s modern geometry Geometry in Dimensions Geometry in Four Dimensions Geometry in high dimensions Complete chart up to the 14th dimension Stereochemistry and homochirality Five manipulations of proportions tesseracts and hypertesseracts Polygons Definition of a polygon Golden rectangles Proofs Proof of a theorem in paragraph form Hypothesis and conclusion Indirect proofs Hunch, hypothesis, theory, and law Proofs of all the area formulas given only the area of a square (This is hard. Most books start with the area of a triangle as given.) Proofs of the Pythagorean theorem Definition of a limit of a function Inductive and deductive reasoning Proofs using geometry Unlike all other math programs, this one also has: • The only verse of Fred’s famous song, “Another Day, Another Ray” • The difference between iambic, trochaic, anapestic and dactyllic in poetry • How easy it is to confuse asinorum which is in the genitive plural in Latin with asinus which is in the nominative singular. • A good use for Prof. Eldwood’s Introduction to the Poetry of Armenia while on the deck of a pirate ship

All answers are included in the textbook.

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