Two polygons, with the same number of sides, are similar when they have angles congruent matches and matching sides proportional. In other words, similar polygons have the same shape, but their dimensions are not always the same size. Note in the image below an example containing two triangles similar. As these figures are also polygons, so this is also your definition of resemblance.
You triangles they are polygons which have the least number of sides, therefore, it is possible to create strategies to reduce the work of checking the resemblance between them. These strategies are known as triangle similarity cases and will be discussed below.
1st case of similarity: Angle-Angle (AA)
whenever two triangles have two angles congruent correspondents, they will already be completely similar. Note that if two triangles have two congruent angles, they also have the third congruent angle. This is guaranteed by the sum of the interior angles of the triangles, which will always equal 180°.
The following example shows in red two congruent angles of two.
triangles distinct. The rest of the measurements were grayed out just to see the resemblancein between you triangles.
Note that the corresponding sides of these two triangles are proportional and that the remaining angles, highlighted in gray, are congruent.
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2nd case of similarity: Side-Side-Side (LLL)
Whenever two triangles have three corresponding proportional sides, then they will besimilar. In other words, triangles that have three proportional sides always have the corresponding congruent angles.
The following example shows two trianglessimilar, as they have the measures of their three proportional sides. In gray are the measurements of the angles of these triangles.
3rd case of similarity: Side-Angle-Side (LAL)
If two distinct triangles have two proportional sides and the angle between those sides is congruent, then these two trianglesthey aresimilar. In the following image, see an example of triangles with two sides proportional and the angle between them congruent. We put the rest of the triangle measurements in gray in the example to show the similarity between them.
Example
Both triangles next are similar. Determine the DF segment measure.
like two trianglessimilar have proportional corresponding sides, to find out the measure of x, just assemble the proportion:
5 = 4
x 14
4x = 5.14
4x = 70
x = 70
4
x = 17.5 cm
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "What is similarity of triangles?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-semelhanca-triangulos.htm. Accessed on June 27, 2021.
