Signs of similar triangles: the concept and scope

important concept in geometry as a science, is the similarity of figures.Knowledge of such properties can solve a great number of tasks, including in real life.


similar figures referred to are those who can be transformed into each other by multiplying all sides by a certain factor.Wherein respective angles should be equal.

consider in more detail the signs of similarity of triangles.There are three rules that allow us to assert that these figures have this property.

The first sign of the similarity of triangles requires the equality of the two pairs of corresponding angles.

According to the second rule, regarded such figures are considered as two sides of one another are proportional to the respective segments.The angles which are formed by them must be equal.

And finally, the third sign: the triangles are similar if all their sides are proportional, respectively.

There are some figures that in some properties can be attributed to a special type (equilateral, isosceles, rectangular).To claim that these triangles are similar, you must perform fewer conditions.We have for example, consider the similarity of the signs of right-angled triangles:

  1. hypotenuse and one of the legs of one proportional to the corresponding sides of the other;
  2. any acute angle of one shape is the same in another.

if you observe signs of similarity of triangles, have the following properties:

  1. ratio of linear elements (medians, bisectors, heights, perimeters) is equal to the similarity;
  2. if you find the result of dividing space, we obtain the square of this number.


above properties allow to solve a great number of geometric problems.They are widely used in life.Knowing the signs of similarity of triangles, you can determine the height of any object or to calculate the distance to the remote point.

To find out, for example, the height of the tree in pre-measured distance vertically mounted pole, which is secured revolving bracket.It is oriented to the top of the object and the mark on the ground the point where the line, continue it intersects a horizontal surface.We get these right triangles.Measure the distance from the point to the pole, and then to the subject, we find similarity coefficient.Knowing the height of the pole, you can easily compute the same parameter for the tree.

To find the distance between two points on the ground plane to choose another one.Then measure the distance from it to available.Connect all the dots on the ground and measure the angles that are adjacent to the famous side.By building on the paper like a triangle and defining the aspect ratio of the two figures, easy to calculate the distance between two points.

Thus, signs of similarity of triangles - one of the most important concepts of geometry.It is widely used not only for scientific purposes, but also for other purposes.