Similarity of Triangles

Similarity theory for triangles is sufficient for high school level  trigonometry. 

Similarity theory in Euclidean geometry may say 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. That being said, 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 theory for maps and plans with coordinates (analytic geometry)  may take a more general viewpoint:   That is, two figures in the plane are similar when and only when after a change of scale if need-be, they have or correspond to the same set of coordinates.  This analytic viewpoint is needed to understand or codify our ability to  recognize letters and objects which have the same shape or nearly the same shape but different sizes in reading and writing letters and symbols, and  in recognizing  objects drawn on paper or as they exist in real life (space).   In the foregoing,  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.

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.   To learn more about similarity, visit  Maps, Plans,  Similarity & Trig, (alt view)

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

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.  

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

are enough to imply 

and hence the similarity of triangles. The case of right triangles leads to trigonometry. 

End Note 1:  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"?

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