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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
Euclidean Geometry LeanlyA description of site treatment of Euclidean Geometry as it is or will be follows. Here will be means a revision of what is online is pending. Site treatment starts off with the definition of a triangle as three vertices in the plane (non-collinear preferred) with the line segments that join each pair. The vertices determine a triangle. Correspondence (pairings) between the sides and angles of two triangles are implied by correspondence or mapping of the vertices of one with the vertices of another. There-in lies an example of a function (mapping, arrow diagram) in geometry. How a transversal between two lines or line segments leads to a correspondence between angles of the intersection of transversal with the line segments (corresponding angles) leads to another function example. Then an algebraic theorem implying the sum of interior angles is 180 degrees is (i) equivalent to the equality of alternate angles, and (ii) equivalent to corresponding angles being equal sets the stage for a later characterization of parallel lines in terms of any one of these equivalent conditions. The site treatment then review side-side-side, side-angle-side and angle-side-angle triangle construction methods, and their limits, that is the conditions where they work or fail. Identification of when the angle-side-angle method fails suggests the parallel postulate, namely that two lines are parallel when and only when the the sum of angles for a transversal between them is 180 degrees (or two right angles). The algebraic theorem gives equivalent conditions for parallelism in terms of equality of alternate angles or corresponding angles for a transversal. A simple proof that the sum of angles in a triangle is 180 degrees then follows using the equality of alternate angles between one side of a triangle and a parallel line through the third side. The assumption that side-side-side, side-angle-side and angle-side-angle triangle construction methods can be used to duplicate a triangle in its original location or another leads to three triangle isometric or congruence conditions, called side-side-side, side-angle-side and angle-side-angle as well. For isoceles triangles, the equality of two angles at two of the triangles vertices implies both are acute, and that appears to restrict the location of the third vertex - it must lie above the base - the line segment joining the two equal angles. Joining the midpoint of the base to the third vertex of the triangle leads to two triangles that are isometric by the side-side-side triangle isometry postulate. Whence sides opposite the two angles are isometric - have equal measures. Conversely, when two adjacent sides in a triangle are equal, the bisector of the angle between them appears to intersect the third side, and so form two triangles that are seen to be isometric by the side-angle-side isometry postulate. Whence angles opposite the equal sides correspond and so must have equal measure by triangle isometry. A convex quadrilateral is given by four vertices joined by line segments in a manner that alternate ones do not cross. Drawing a line segment between the first and third (or second and fourth) vertices determining the quadrilateral appears to give a diagonal. This diagonal (or the oter one) divides the quadrilateral into two triangles. the following conditions are equivalent.
The first condition 1 implies each of the other three as corresponding sides are equal, and the equality of triangle corresponding angles on alternate sides of the diagonal implies the sides of the quadrilateral are parallel. That being said, conditions 2, 3 and 3 imply condition 1 by side-angle-side, side-side-side and angle-side-side triangle isometry or congruency postulates, respectively. In each application, a diagonal provides a common side. The equivalence of the foregoing conditions implies 4 different ways to recognize or construct a parallelogram. When a line is drawn through a triangle in a way that the line is parallel to one side, a smaller triangle appears to result. In the original and smaller triangle, corresponding angles are equal. Moreover, measurement suggests that corresponding sides are proportional. For an pair of triangles, the similarity postulate assumes that (1) corresponding angles are equal when and only when (2) corresponding sides have proportional lengths. Two triangles are said to be similar when and only when (1) or (2) holds for some correspondence between their vertices. |
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