MHB Why do similar triangles have equal ratios of sides?

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SUMMARY

Similar triangles exhibit equal ratios of corresponding sides due to their equiangular properties. This relationship is foundational in trigonometry, where the fixed values of sine and cosine functions for specific angles are derived from these ratios. The slope of a straight line, represented by the tangent of the angle it forms with the positive x-axis, further illustrates this concept. Understanding these relationships allows for straightforward proofs of the properties of similar triangles.

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  • Understanding of basic triangle properties and definitions
  • Familiarity with trigonometric functions: sine, cosine, and tangent
  • Knowledge of equiangular triangles and their properties
  • Basic geometry concepts related to angles and slopes
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  • Study the proof of equiangular triangles being similar using ProofWiki
  • Explore the properties of trigonometric functions in relation to angles
  • Learn about the relationship between slopes and angles in coordinate geometry
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Amer
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I was wondering why similar triangles have the property that the ratios of the similar sides are equal.
Or why the triangular functions (sin, cos,...) for a certain angle is fixed.
They are related, and if I can find one of them, the other can be proved easily.
I was thinking about the slope of the straight line since it is fixed and it is equal to the tan(angle which the line made with the positive x-axis) .

Any ideas?

Thanks.
 
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... You can just prove buy saying that angle AB,for example, is equal to AB2
 
Amer said:
I was wondering why similar triangles have the property that the ratios of the similar sides are equal.

Hi Amer, :)

Refer either of the following links.

1) Equiangular Triangles are Similar - ProofWiki

2) http://farside.ph.utexas.edu/euclid/Elements.pdf (Page 160)
 

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