MHB Why do similar triangles have equal ratios of sides?

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Similar triangles maintain equal ratios of corresponding sides due to their equiangular nature, meaning their angles are congruent. This property leads to proportional relationships among the sides, which can be demonstrated through geometric proofs. The discussion also connects this concept to trigonometric functions, noting that the sine and cosine values for a given angle remain constant, reflecting the fixed ratios in similar triangles. The slope of a line, represented by the tangent of the angle it forms with the x-axis, further illustrates these relationships. Understanding these connections enhances comprehension of both triangle similarity and trigonometric principles.
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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)
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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