Similar triangles

In Euclidean geometry, two objects are similar if they are the same shape, regardless of their size.

Two triangles are similar if, and only if, their angles are congruent (equal) and their corresponding sides are proportional (homologous). That is to say, two triangles are similar if the only difference is size.

Homologous sides are those opposite at equal angles.

Here we have a example, where the homologous elements (angles and sides) are seen with the equality or congruence of their angles and the proportionality of their sides:

In similar triangles the following conditions are met:

• Homologous angles have the same measure (AAA Theorem). That is:
  
• Homologous sides are proportional, that is, the lengths of the corresponding sides have all the same ratio (SSS Theorem). For instance:
  

  The ratio r of corresponding sides is called scale factor.

• The perimeters’ ratio of two similar triangles is also their scale factor. So, the perimeters of the two similar triangles are in the same ratio, as the sides. Moreover, their areas’ ratio is equal to the square of the scale factor:
  

To determine if two triangles are similar, it is not necessary to know their three angles and their three sides. Remember that we know if two triangles are similar when their corresponding angles are congruent, and their corresponding sides are proportional. There are three statements, or theorems, each of which is necessary and sufficient for two triangles to be similar:

Triangle Similarity Theorems

1. AA Theorem. The triangles have two congruent angles. The third angle will also be because they have to add 180°.
   

   If α = α’, and β = β’, then ABC and A’B’C’ are similar.

   Congruent angles criteria:

   • Homologous sides are parallel lines (see picture above).
   • The three sides of a triangle are perpendicular to the homologous sides of the other triangle.
     
2. SAS Theorem. The two triangles have two proportional sides and the angle between them is equal.
   

   Then:

   

   Therefore, ABC and A’B’C’ are congruent.

3. SSS Theorem. Two triangles have their three sides proportionals.
   

   Then:

   

   Thus, ABC and A’B’C’are similar.

Triangles in Thales Position

Two triangles are in Thales position if they have a common angle, and they have parallel opposite sides. As with the rest of the polygons, if two triangles can be placed in Thales position then they are similar.

This statement is the one that establishes the Intercept theorem (also called Thales’ theorem). So, the Intercept theorem can be used to prove the properties of similar triangles.

Then, it is true that:

Exercise 1

Given two triangles with all of their sides measures, determine whether the triangles are similar or not.

Lengths of the sides of the first triangle: 7.6 cm, 4.18 cm, and 6.65 cm.

Lengths of the sides of the second triangle: 4 cm, 2.2 cm, and 3.5 cm.

SOLUTION:

Since we know the lengths of the triangle’s sides, we can apply the Side-Side-Side (SSS) Similarity Theorem:

Then, we can determine the scale factor and find out if all sides are proportionals.

As we see, the ratio between the corresponding sides of the two triangles is the same (similarity ratio or scale factor = 1.9). So, the two triangles are similar.

Exercise 2

We have two triangles: the larger one, two sides of 10 cm and 5.5 cm concur in the angle γ of 70°, while the smaller one has three sides, 4 cm, 2.2 cm and 3.5 cm. Determine if these triangles are similar.

SOLUTION:

In this instance, the three known data of each triangle do not correspond to the same criterion of the three exposed above. To find the unknown side c in the larger triangle, we resort to the procedure set forth in solving triangles, in the section on “knowing two sides and the angle they form”, in which we must apply the Law of Cosines.

Apply the Cosine Rule:

Side c is 9.64 cm.

Since we already know the length of the three sides of each triangle, we can calculate the ratio between each pair of homologous sides.

We use the SSS Theorem,

And then, we check that the ratios are not all the same. The ratios are 2.5, 2.5 and 2.75. Therefore, these two triangles are not similar.