Why do similar triangles have equal ratios of sides?

In summary, similar triangles have the property that the ratios of the similar sides are equal. This is because they have the same angles and therefore the same ratios between their corresponding sides. This is also related to the triangular functions (sin, cos,...) for a certain angle being fixed, as finding one of them allows for the other to be proved easily. The slope of a straight line, for example, is equal to the tan(angle which the line made with the positive x-axis), which can be used to prove the equality of angles in similar triangles.
  • #1
Amer
259
0
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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  • #2
... You can just prove buy saying that angle AB,for example, is equal to AB2
 
  • #3
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)
 

FAQ: Why do similar triangles have equal ratios of sides?

What is a "Similar triangles Proof"?

A "Similar triangles Proof" is a mathematical proof that shows that two triangles are similar, meaning that they have the same shape but possibly different sizes. This is usually done by proving that their corresponding angles are equal and their corresponding sides are in proportion.

Why is it important to prove that triangles are similar?

Proving that triangles are similar allows us to use the properties of similar triangles in solving problems. These properties include the fact that corresponding angles are equal, corresponding sides are in proportion, and the ratio between the sides is constant. This can help us find missing side lengths, angles, and solve real-world problems.

What is the process for proving that triangles are similar?

The process for proving that triangles are similar involves showing that their corresponding angles are equal and their corresponding sides are in proportion. This is typically done using theorems and postulates, such as the Side-Angle-Side (SAS) or Angle-Angle (AA) similarity theorems. The proof may also involve using algebraic equations to show that the ratios between corresponding sides are equal.

Can you give an example of a "Similar triangles Proof"?

One example of a "Similar triangles Proof" is proving that two triangles are similar using the Side-Angle-Side (SAS) theorem. This involves showing that two sides of one triangle are in proportion to two sides of the other triangle, and that the included angles are equal. By doing so, we can conclude that the two triangles are similar.

What are some real-world applications of "Similar triangles Proof"?

"Similar triangles Proof" has many real-world applications, such as in architecture, engineering, and map-making. It is also used in various fields of science, such as in physics to calculate distances and in biology to study the similarities between different species. In the real world, similar triangles can be used to find the height of a building, determine the size of a shadow, or even estimate the size of an object from a distance.

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