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Geometry in 15 Steps

Most later topics have no take-home value besides preparation for calculus-based college programs in business, science, technology, and further mathematical subjects.

Notes for Instructors

Maps and plans of places and building help people with navigation. Measurement of angles and scaled lengths on maps and plans, and the location position on maps and plans provide further examples of geometry useful not only in navigation but also in decorating and building activities at home and at work. Thus some geometry with maps and plans may have take home value. The evaluation of geometric formulas for perimeters, areass and volumes has been included with site algebra starter lessons instead of here.

The geometry material here may serve secondary and college students, say 13 to adult. The objective here is to provide an confidence-building, practice-first, theory second, operational command of skills and concepts because most students who might follow the paths below will not be heading into pure mathematics. Skill development does not require a full recognition of all the assumption implicit in practice. Students need to learn to reason mathematically to the maximum extent possible, but with empirical checks on physical implications. An element of doubt may encourage that. The treatment below could be enriched by explaining how reasoning with sketches drawn with insufficient precision may fail, a failure that can be explained via the use of coordiantes.

The main points of the material follow.

Triangle construction and triangle isometry criteria, two sides of the same coin, with applications to drawing perpendiculars and bisectors, develop the plane geometry abilities of students. Their construction of paper models or furniture to scale or full size may develop measurement skills and abilities.
Learning how to use rulers protractors and drawing instruments are observable skills that check or corrected.
Rectangular coordinates and polar coordinates allows translations, rotations and reflection to given and described with order pairs. That implies a very simple geometric introduction to complex numbers which can be employed to consolidate student command of the law of signs, and to demystify the square roots of negative numbers.
The first treatment of Lines and Slopes assumes the ability to solve linear equations - Site starter lessons may help with that. For given rotation, This treatment observes that the midpoint between two points is rotated into the midpoint of the images under rotation of the two points.
While the right triangle development of trigonometry may begin with assumptions about similarity of triangles, the more general analytic view of similarity reflects the similarity of man-made objects with their designs on paper or in inside a computer file. The question of what is similarity provides motivation for the more general view - motivation that students heading for college programs in science, technology, engineering and mathematics should meet.
The coverage of complex numbers goes beyond the complex number appetizer present in the discussion of rectangular and polar coordinates. When a triangle is rotated about one of it vertices, the midpoint of the opposite side is rotated into the midpoint of the rotated side. That together with the observation that a similarity transformation commututes with a rotation implies the distributive law for complex numbers. Whence all the field or arithmetic properties of the complex numbers are consequences of the corresponding properties of real numbers, and the distributive law. In the 1950s, the modern mathematics course designers choose to give students axioms for real numbers and not complex numbers. The latter axioms would have made the high school coverage of regular polygons and unit-circle, periodic, trigonometry functions very simple. That simplicity is implied here due to a very simple, high school school derivation of the distributive law for complex numbers.
The UK development of slopes in 1967 came after an introduction of the tangent function. The second development of straight lines and their slopes and equations makes the link between slopes and the tangent of angles of inclination.
The geometric subject of how lines parallel to the base of triangle cut the remaining two sides in a proportionate manner is covered here under the topic of how lines parallel or instersecting are cut in proportionate ways by parallel transverals.

maps and plans to trigonometry and vectors

Maps Plans Measurement:
describes length measuring devices, direct area calculation, length and area calculation from maps and plans, the preservation of angles on maps and plans drawn to scale, the distortion of angles and proportions on maps and plans not drawn to scale, and the use of maps and diagrams not drawn to scale. Here are skills to check via questions and exercises.

Euclidean Geometry - Constructions Theory extras: covers the concepts of points, line segments, lines and rays; the meaning of correspondence between triangles or their vertices; the isometry and congruence of triangles; ruler and compass constructions for triangles, angle bisection and dropping a perpendiclar; and what is an isoceles triangle.

A discussion of when triangle construction fail sets the stage for the parallel postualate and then an explanation of why angles in a triangle sum to 180 degrees or two right angles. Lessons on angles subtended by chords in circles and on circles related to the right bisectors of triangle sides follow. Preparation for trigonometry is provided by a discussion of similarity of triangles in general and then for right triangles. The discussion of kite and parallelogram construction - a site add-on [?] for Euclidean Geometry - set the stage for vectors and for the site and for site vectorial views or development of the operations on real numbers. The coverage of polar and Cartesian coordinates, operations with and distributivity laws sets the stage for complex numbers, and the development of its properties without the use of the Pythagorean theorem in their derivation. A site content shuffle may move these add-ons or extras in a folder separate from the treatment of Euclidean Geometry.

Cartesian and Polar Coordinates: introduces or reviews the use of these coordinates along number lines, in a plane and in space for the location of points. The absolute value and Pythagorean calculation of distance between numbers or points are introduced. Included here is the Chinese square dissection proof of the Pythagorean - a proof that mixes geometry and algebra. A simple complex number appetizer or starter lesson is based on adding with Cartesian or rectangular coordinates and multiplying with polar coordinates. These operations are easily linked to translations and rotation in the plane, and a geometric view of the law of signs for real numbers. The full treatment of complex numbers goes futher - it includes a very simple mid-point rotation proof of the distributive law for complex multiplication over complex addition.

The triangle inequality is explained for points in a line and in the plane. In the case of the number line, this inequality is connected to the addition and subtraction of collinear arrows with the same or opposite directions.

Here and elsewhere, the terms Cartesian coordinates and rectangular coordinates are equivalent - interchangeable.

Lines and Slopes Take 1: develop formulas for slopes, for the point slope equation of line, for two further forms of equations. for the product of slopes of orthogonal oblique lines, and for locating the intersection of non-parallel lines. The latter requires exercises student skills in solving linear equations in two unknowns. The treatment here emphasizes numerical skill and the geometric-algebraic development of of results and formulas gives students a chance to test or improve their algebraic reasoning skills. That is best done sooner instead of later for students aiming for college programs in business and technical fields.

Included here are the development of formulas for the mid-point of a line segment. Midpoint formulas and properties will be employed later in the graphing of inverse function and, as indicated above, in the full treatment of complex numbers.

This first treatment of lines and slopes avoid mention of angles of inclination and how the slope of a line is equal to the tangent of that angle. That perspective and its consequences is left to the second treatment of lines and slopes, a treatment that includes or reflects the unit circle development of the tangent function. The symmetric form of an equation for a straight line is not yet covered in this folder.

What is Similarity: provides a general and unified treatment of the likeness or similarity of squares, circles, triangles and arbitary regions in the plane. The lessons here can be covered after section 1 on maps, plans and measurement and after the introduction of Cartesian coordinates. The objective here is to explain and reconcile different characterizations of similarity.

The key question is how to recognized similarity. In modern life, objects are similar by design if they stem from the same plans but are built to different scales. That reflects a coordinate view point of similarity objects. Two objects are similar if we can attach coordinate system to each so that the set of coordinates for the points for one object essentially provides the plan for the other as is or after the application of a scale factor - a dilatation. This set of coordinate development easily explains how and why circles, squares and rectangles - those with a common aspect ratio - may be similar. The coordinate perspective of similarity in the case of similar triangles and more generally in the case of similar polygons implies corresponding angles are equal and corresponding sides are proportional. A partial proof of the converse is included in the section what is similarity - a full proof is left to later as site development to do.

Trigonometry first steps: introduces trigonometric ratios for right triangles and shows that for a given acute angle, those ratios are independent of the scale at which a right triangle with the acute angle is drawn. The property that the sum of angles sums to 180 degrees in a right angle is employed here. That property is proven in the site section on Euclidean Geometry - the coordinate-free development; and also in the section below on Parallel Straight lines and transversals - a coordinate based development. Lessons 3 to 5 say how to calculate sines, cosines and tangent ratios or functions for acute angles using right triangles. The values of these ratios or functions for angles 45, 30 and 60 degrees are found from two special families of similar triangles (i) isoceles right triangle with angles 45-45- 90 degrees, and (ii) the half-equalateral right triangle with angles 30-60-degrees. The two triangles are employed to obtained exact arithmetic expressions with surds, that is square roots of 2 and 3, in place of decimals. Preparation for college programs require this non-decimal, exact viewpoint. The section begins with my view, the whyslopes explanation, of why trigonometry is employed, and how drawing diagrams to solve missing lengths and angles geometric problems would be a viable alternative, but for the requirement of college programs in mathematical disciplines - see Postscript C in the site folder Euclidean Geometry.

Complex Numbers:
reproduces and continues the complex number appetizer or starter lesson in the section above on Cartesian and Polar Coordinates. It adds easy rotate a midpoint proof of the distributive law for complex numbers. All other algebraically described, field properties of complex numbers are consequences of the algebraically described, field properties of real numbers, alone or in combination. The site development of the mid-point formula depends on the Pythagorean theorem.

Site Volume 3, Why Slopes and More Mathematics, includes a proof of the distributive law that depends on similarity properties and not the Pythagorean theorem. Whence two ways to calculate the product of a complex number a + ib of modulus r with its complex conjugate a - ib implies r² = a² + b² gives another proof of the Pythagorean. That being said, the latter proof with the easy midpoint based development of the distributive law, becomes a confirmation.

The development of complex numbers before or besides the introduction of unit-circle definition of circular or periodic trigonometric functions permits the use of complex number properties and techniques in the derivation and justification of trigonometric formulas. Lessons provides examaples in the form of trignometric angle-sum, double-angle and triple angle formulas. Further more, trigonometric formulas for the dot- and cross-product expressions that appear in two dimensional vector analysis, the coordinate development, follow easily from complex number considerations. The cosine law for scalene triangles, one vertex at the origin, is implied by the trigonometric formula for dot products in the plane. A converse to the Pythagorean is an immediate and easy consequence of the cosine law. See lessons 8 to 15.

Cube, sixth and N-roots of unity and N-roots of complex numbers are described in lessons 17, 18 and 19. The presentation consists of hand-written, poorly recorded on a pen tablet apart from a computer, and then uploaded. So the writing and presentation is not optimal. None the less, teachers and tutors may recognize the essential ideas and present them in class. The discusion of N-th roots of unity connects complex numbers to regular N-gons in the plane. Complex- and real-number for calculating N-th roots are compared and contrasted in lesson 20.

The discussion of logarithms, powers and exponentials in lesson 21, a reproduction of the last chapter in site Volume 3, Why Slopes and More Mathematics, gives a formula based approach that extends the definition of - the description of how to compute - logarithms, powers and exponentials from the case of real numbers to case of complex numbers. This discussion is too deep for high school studies. It may however serve as appetizer for undergraduate courses in advanced mathematics.

Unit-Circle Trigonometry: mostly comes from hand-written, poorly recorded on a pen tablet apart from a computer, and then uploaded. So the writing and presentation is not optimal. None the less, there is wide but not full treatment here of circular trigonometric functions. Lessons 17A to 17G duplicate lessons in the Complex Number made easy folder to derive trigonometric identities and formulas. Lessons 32 to 35 also employ complex number techniques to derive multiple angle formulas for sines and cosines of angles 2A, 3A, 4A and 5A.

Lines and Slopes Take 2 with tangent function: relates slope of straight lines to the tangent of angle of inclination. To that end, this folder includes a development of the tangent function independent of the development of trigonometric ratios and functions in the other sections. With that this development of slopes and lines, and their equations is independent of the earlier development as well. Lessons 8 to 14 may provide a second or first perspective of lines, their slopes and equations. That being said, the intersection of straight lines is not considered in this folder. The treatment in Slopes and Lines Take 1 could be covered here.

Intersecting Straight Lines and Transversals: extends the earlier coverage of intersecting straight lines. The extension includes an analytic view of triangle construction methods - Angle Side Angle and Side Angle Side. The lessons or notes are simply the product of curiousity - an endeavour to see what might be included or not in future course design and delivery. The content may enrich studies of gifted students - students struggling to master routine material should avoid this folder.

Parallel Straight Lines and Transversals: provides an analtyic geometry, coordinate-based development of parallel straight lines and their properties. Included here is analytic proof that the sum of angles in a triangle should be 180 degrees. The second lesson on the proportionality of line segments generated by a parallel transversals may read besides lesson elsewhere on properties of triangles formed within another by lines parallel to one side.

Function Translating and Rescaling: describes the relation between the graph of a function y = f(x) and the graphs of functions y = Af(x) and y=Af(x-c)+K in some special or general cases. In retrospect, the coverage of this topic is partial and could be extended.

The 13 Vectors:
folder includes a simple geometric introduction with vectors first seen as arrows indicating movements on a map, with head-to-tail addition of successive movement being easily defined and clearly having an associative property. The subsequent representation of 2D movements or vectors with coordinates implies further properties of vector addition, properties which reflect those of real numbers. That implication completes a one revolution of a spiral, if not a circle in that the site number theory lessons include a derivation of properties of real and rational numbers from their identification with 1D collinear movements in the same or opposite directions.

This vector folder or section reproduces the complex number development of trigonometric formulas dot- and cross-products. In particular, the dot product of two vectors is seen to be the product of the lengths and the cosine of the angle between them. Students who study linear algebra may obtain the same result derived for non-collinear vectors in 3D by applying the Gram-Shmidt process to obtain an orthogonal base for IR³.

Degrees to Radians and Radians to Degrees: is a technical topic employed in the discussion of circular motion in physics (correct me if I am wrong) at the high school level, and employed in calculus the development of slope or derivative formulas for the nonlinear sine and cosine. Calculus is the subject of slope related computations and interpretations, forwards and backwards. In it formulas for slopes to nonlinear functions y = f(x) are derived from formulas for y =f(x). Whence the term derivatives appear. The conversion of angles between degrees and radians is part of the mastery of radian measure for angles.

Arc or Inverse Trigonometric Function: employs the function concepts of inverse function and domain restriction to specify intervals on which trigonometric functions are one-to-one, and then to defined inverses to the restriction of trigonometric functions to those intervals. More than one choice of the interval is possible. Thus different texts and courses may employ slightly different definitions for inverse trigonometric function. Lessons here why the inverse of sine, cosine and tangent are called arcsine, arccosine and arctangent. Inverse trignometric functions are typically employed informally in the earlier "triangle solving phase" of trigonometry. The discussion here may begin before a first course in calculus and be completed during and first or second course of calculus.