www.whyslopes.com   << Français :  20 pages >>    
Appetizers and Lessons for Mathematics and Reason
  online logic chapters  - the best starting point for further site exploration.  Bon Appetite.

Similarity
Back ] Home ] Next ]


Euclidean Geometry
(Essential Elements)

Similarity

Back ] Home ] Next ]

What is Correspondence
Isometry
Side-Side-Side
Bisecting Angles
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

 Deductive logic in maths may begin
  here.  But deductive   logic mastery 
itself  may begin with words and stories.

Complex Numbers
Update (December 13th, 2009). A
simpler & quicker development of 
complex numbers  is  available: 

 


In reading and writing letters and in seeing objects as they exist or on paper, we recognize letters and objects which have the same shape or nearly the same shape but different sizes. Size but not shape varies as we move to or away from the letters or object. Size thus depends on distance. The geometric theory of optics says or suggest how. 

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.

Digression: The Site Reviews  
  • Magellan, the McKinley Internet Directory, 1996: Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.   (Magellan is no longer online)
  • The World-Wide Web Virtual Library Education by Country - Canada 1, 2005. Why Slopes: Appetizers and Lessons for Math and Reason. This online classroom offers appetizers and lessons for math from arithmetic to calculus or why slopes; for deductive reason (logic) and critical thinking; and for learning in general. Included here are opinions on the communication of skills and mathematics instruction. The logic appetizers are math free. Each appetizer is different. If one is not to your liking try another. Most are from three books on understanding and explaining math and reason.

may encourage a visit to the site entrance www.whyslopes.com. Bon Appetite.

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

  1. The ratios of  lengths of corresponding sides are all equal. That is
    a
    d
     =  b
    e
     = c
    f
  2. 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  ABC is part of a larger figure, or when the angle at vertex A is not unique, we may write  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
f

as 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
b

 = 

d
e

a
c

 = 

d
f

b
c

 =

e
f

are enough to imply 

a
d

 = 

b
e

 =

c
f

 =    

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

site search

Parents: Help your Child/Teen Learn covers  Speaking Skills, Reading & Writing Preparing for Science Having Patience, etc

Math How-TOs
1. Arithmetic   2. Algebra   3.  More Algebra  4.  Geometry 5 More Geometry 6.  Calculus
>> densely written 
>> use as skill checklists

Online Volumes (orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3 .Why.Slopes.&
.More.Math.1995

Skill & Concept 
Review or  Development 

 1. Decimal Arith - Video Based ]
2   Fractions  
3.  Fractions  with Units  
3. Solving Linear Equations  - 
making alg easier
4. Formulas forwards & Backwards - unifying theme for Algebra
5.  Proportionality, Back- & For-wards - theme at work.
6.  Logic - Math Free, good for precision in  work & studies 
7. Euclidean-Geometry  (leanly)
8. Slopes and Lines 
9. Why Study Slopes - a context 
10.  Quadratics
11  Polynomials
12  Factored Polys - a context
13 Functions - For-& Back -wards
14  Number Theory, Richly
15. Exponents, Radicals & logs.  
16   Calculus - Examples & Advice 
17.   Real  Analysis 
18  Electric Circuits Etc (So So)
19 Maps, Similarity & Trig, (alt view)
20 Complex numbers  

21 Logic with Symbols+truth tables

22  Consistent Story Telling
23. Even More Logic

 Back ] Next ] [Top of this Page]  
  www.whyslopes.com?

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by
email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby
,  All Rights Reserved.