YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work |
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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
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
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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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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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.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
Similarity theory in geometry says when two polygonal figures have the same shape. The latter codifies the notion of two planar regions or curves in plane having the same shape, incompletely as only polygonal curves and regions are considered, but that is good enough for the further needs of high school mathematics. The further study of trigonometry with triangles is based on the similarity of right triangles. Similarity theory for triangles would be sufficient for high school mathematics - see trigonometry. Two planar polygonal figures (triangles) are similar when and only when (i) corresponding angles are equal (have the same measure) and (ii) corresponding sides are proportional. Trigonometry on or with the unit circle provide another face of the further study of trigonometry. |
Similarity of Triangles
Two triangles 
ABC and DEF
are said to be similar with respect to a correspondence when and only when both of the following conditions are satisfied
- The ratios of lengths of corresponding sides are all equal. That is
a
d= b
e= c
f - Corresponding angles are equal.
The shorthand notation. ABC 
DEF means the two triangle are (or
should be) similar.
The empirical pattern that appear after drawing and measuring many triangles suggests the following:
Assumption: The conditions (1) and (2) are equivalent. That is, if one of them holds, so must the other.
Therefore in order to decide whether or not two triangles are similar, we only have to check whether or not (1) ratios of corresponding sides are equal; or (2) corresponding angles are equal. Checking only one of the two conditions leads to minimal conditions for similarity.
Minimal Condition AA. The proof (given later) that the sum of angles in a triangle is two right angles (180 degrees) implies that if a pair of angles in one triangle coincide with a pair of angles in a second triangle, then the third angles are equal too. Hence there is a correspondence in which matching (that is corresponding) angles are equal. (The proof that three angles in a triangle sum to 180 degrees depends on the properties of parallel lines.)
Remark 1. For two triangles to be similar, the longest sides and angles must be paired, the smallest sides and smallest angles must be paired, and the other sides and angles must be paired. Pairing here is by the correspondence of the triangles or their vertices.
Remark 2. For clarity, when
CAB in place of
A
Proportionality and Proportionality Constants.
Suppose the triangles ABC and DEF
are similar. Let K be the common value of the ratios or fractions
| a d |
= | b e |
= | c f |
The sides of the first triangle ABC
(lengths in the numerator or on top) are proportional to the sides of the second
triangle DEF (lengths in denominators
or on bottoms) are proportional to the sides
(3) a = Kd, b= Ke and c = Kf
with the common value K as the a proportionality constant. On the other hand (conversely) if equation (3) holds for some constant K, then condition (1) holds.
Proof:
a
d= K and b
e= K and c
f= K Therefore
a
d= b
e= c
fas all are = K
In a typical application of the equations (3), the length of two sides, say a and d serves to find the value of the proportionality constant K. Then as K becomes known, each of the equations b= Ke and c = Kf can be used to find missing lengths when at least one of the lengths in them is known.
| Assumption: The Side-Side-Side Minimal Condition for Similarity.
If there is a matching such that corresponding sides in a pair of triangles are proportional, then the triangles are similar.
|
| The SAS (minimal) condition for similarity assumes if a pair of
matching sides in a pair of triangles have proportional lengths and
their included angles are equal then the pair of triangles are
similar:
The proof of depends on what has been done so far and properties of parallel lines. Another proof follows from the cosine law in coordinate view of trigonometry. |
Remark 3. Let K-1 = 1/K be the reciprocal or multiplicative inverse of K. The the proportionality condition (3) is equivalent to proportionality condition
(4) d = (K-1)a, e =(K-1)b and f = (K-1)c
Here the proportionality constant is replaced by its reciprocal K-1 = 1/K when we go from the second triangle to the first.
Remark 4. Any of two of the three equalities
|
a |
= |
d |
|
a |
= |
d |
|
b |
= |
e |
are enough to imply
|
a |
= |
b |
= |
c |
= |
a |
common |
value |
K |
and hence the similarity of triangles. The case of right triangles leads to
trigonometry.
Scale Factors
for maps and diagrams (and between them)
For triangles (and more generally polygons) in the plane drawn to scale K in maps and diagrams, the scale factor K (for instance 1: 100) provides the proportionality constant. Lengths in the drawing are K times corresponding lengths in the original. Moreover, a topic for later study, areas in the drawing are K2 times those in the original.
Remark 5, A Chain Rule. Suppose we have three drawings, a first, a second and a third, so that lengths in the third are K32 times those in the second, and lengths in the second are K21 times those in the first. Then we may conclude that lengths in the third are K31 = (K32 )(K21 ) times those in the first.
Note according to this author. the "chain rule" for proportionality relations y = a x and z = b y is z = (ba) x. The multiplication of proportionality constants with chains may be associated with models of pulley systems based on ropes or chains, or bicycle gears systems, in which gear or pulley ratios serve as proportionality constants, and coupling of pulley or gears leads to their multiplication. Question: Does this association imply the correct historical origin of the phrase "chain rule"?
www.whyslopes.com
4. Euclidean Geometry
Advice & Directions Correspondence Isometry Side-Side-Side Side Angle Side Angle-Side-Angle Isoceles Right Bisector Construction, Etc. Perpendicular - Point to Line SSS Failure SAS Failure ASA Failure Parallel Lines Angle Sum Similarity Right Triangle Similarity Trig or Similarity Parallelograms Kites From Triangles Duplication Parallelogram from Triangle Duplication Addition of points in the plane Multiplication of Points in the Plane Distributive Law, Step I Distributive Law, Step II Distributive Law, Step III
Above Average Students in Geometry may enjoy the site geometric introduction of complex numbers and the wordy volume 1A, Pattern Based Reason.
For algebra, logic starter lessons, see Volume 2, chapters 1 to 12, plus 14, 16 and 17.
Analytic Geometry, Functions & Trig
(FN) What are Functions?
(FN) Functions - More
SZM: Sign, Zero, Monoticity
(L) Lines Summary
(P) Polynomials (*,+,-)
(Q) Quadratics
(D) Simplify Square Roots
(T) Unit Circle Trig
Conic Sections
More For Analytic Geometry:
Real Numbers
Say More Positive
Linear Inequalities
Triangle Inequality
Absolute Value |x|
|x| Eq'ns & Inequalities
Rectangular Coords
Shortest Path
Distance Formulas
Add & Multiply Points
Polar Coordinates
Radians
(A) Vectors
(A) Coordinate Arithmetic
(A) Navigation on Maps
(A) Addition Geometrically
(A) Rotation
(PT) Translations
(PT) Dilatations
PT: Rotations Easy Consequences of this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint Lesson Plans and lessons
Secondary I - fractions & allied concepts (decimals, percentages)Secondary IV - Functions to Trig & Statistics
Algebra Lesson Notes - All levels
Great_Expectations: If you can learn to follow a multi-step methods in any subject precisely, you can do so in other subjects, as well.
Good news: Site pages identify what you need to study.
Bad news: Site pages do not explain everything
Worse news: Learning takes time, yours