Similarity of Triangles
Similarity theory for triangles is sufficient for high school level
trigonometry.
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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.
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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
- The ratios of lengths of corresponding sides are all equal. That
is
- 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
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= K
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and
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b
e
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= K
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and
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c
f
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= K
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Therefore
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.
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.
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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.
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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:
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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
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a
d
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=
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b
e
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=
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c
f
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=
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a
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common
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value
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K
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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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