Similar triangles are the triangles that have the same shape, but their size may vary. The examples are all equilateral triangles and squares. Similar triangles are different from congruent triangles. In the following sections, we have provided the definition, properties, formulas and theorems of similar triangles. Also, obtain how to find similar triangles and differences between congruence and similar triangles.

Similar Triangle Definition

Two triangles are said to be similar if their corresponding angles, corresponding sides are in the same ratio or proportion. Similar triangles have different lengths of sides but their angles should be equal and their corresponding ratio of the length of the sides must be the same. If two triangles are similar, then it should have

• All corresponding sides of the triangle are proportional.
• All corresponding angle pairs are equal.

Example:

All equilateral triangles are the best examples of similar triangles. The following show that, here you should observe that their sizes are different and shape is the same.

To say that the above two triangles are similar, we have to follow the two important conditions. They are all corresponding sides of the triangle must be similar and corresponding angles are equal.

In △ABC and △DEF

• ∠A = ∠D, ∠B = ∠E and ∠C = ∠F
• $$\frac { AB }{ DE }$$ = $$\frac { BC }{ EF }$$ = $$\frac { AC }{ DF }$$

Therefore, △ABC ∼ △DEF

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Similar Triangles Properties

The important properties of similar triangles are mentioned here:

• Similar triangles have the same shape but different sizes.
• Corresponding sides of similar triangles are in the same ratio.
• In similar triangles, corresponding angles must be equal.
• The ratio of areas of similar triangles is the same as the ratio of the square of any pair of the corresponding sides.

Theorems on Similar Triangles

We can find out whether two triangles are similar or not using these similarity theorems. The three major types of similarity rules, given here.

• AA or AAA or Angle-Angle Similarity Criterion: If any two angles of one triangle are equal to the corresponding angles of another triangle, then those two triangles are similar to each other.
• SAS or Side-Angle-Side Similarity Criterion: If two sides of the triangle are in the same proportion as the two sides of another triangle, and the angle inscribed by two sides in the triangle are equal, then two triangles are similar.
• SSS or Side-Side-Side Similarity Criterion: If all three sides of the triangle are in proportion to the three sides of another triangle, then those are similar.

How to Find Similar Triangles?

Two triangles can be proved similar by using the similarity theorems. And make use of these steps to check if the given triangles are similar or not.

• Note down the given dimensions of the triangles.
• Check if these follow any of the criteria for similar triangles theorems (AA, SSS, SAS).
• If they satisfy any of the similarity theorems, then place similarity symbol between them.

Similar Triangles Vs Congruent Trinagles

These are the differences between congruent and similar triangles.

Similar Triangles Congruent Triangles
These triangles have the same shape but are different in size. These triangles are the same in size and shape.
It is represented by the symbol ‘~’. It is represented by the symbol ‘≅’.
The ratio of all corresponding sides is equal. The common ratio is called the scale factor. The ratio of corresponding sides is a constant value.

Solved Questions on Similar Triangles

Question 1:
Check if two triangles are similar.

Solution:
The sum of all internal angles in a trinagle is 180 degrees.
In triangle ABC,
∠A + ∠B + ∠C = 180°
∠A + 70° + 50° = 180°
∠A = 180° – 120°
= 60°
In triangle XYZ
∠X + ∠Y + ∠Z = 180°
60° + ∠Y + 50° = 180°
∠Y = 180° – 110°
= 70°
So, ∠B = ∠Y and ∠A = ∠X
Using AA rule, ΔABC ~ ΔXYZ.

Question 2:
In the ΔABC length of the sides are AD = 10 cm, BD = 25 cm and BC = 30 cm. Also, DE||BC. Find DE.

Solution:
In ΔABC and ΔADE, ∠DAE is common and ∠ADE = ∠ABC (corresponding angles)
ΔABC ~ ΔADE using AA Criterion
$$\frac { AD }{ AB }$$ = $$\frac { DE }{ BC }$$
$$\frac { 10 }{ 35 }$$ = $$\frac { DE }{ 30 }$$
$$\frac { 2 }{ 7 }$$ = $$\frac { DE }{ 30 }$$
DE = 10 cm

1. What are three similar triangle theorems?

The three similarity theorems are AAA or Angle-Angle Similarity theorem, SAS or Side-Angle-Side Similarity theorem, SSS or Side-Side-Side similarity theorem.

2. Which type of triangles are always similar?

Equilateral triangles are always similar. Any two equilateral triangles are always similar irrespective of the length of the sides of the equilateral triangle.

3. What are the rules for similar triangles?

The two important rules for the two triangles that are similar are mentioned here. All corresponding angle pairs of triangles should be the same. All corresponding sides of triangles are in the same proportion.

4. What is the formula for similar triangles?

If all three sides of the first triangle are in proportion to the three sides of another triangle, then two triangles are similar. If $$\frac { AB }{ DE }$$ = $$\frac { BC }{ EF }$$ = $$\frac { AC }{ DF }$$, then △ABC ~ △DEF.

All the corresponding angles should be equal. If ∠A = ∠D, ∠B = ∠E and ∠C = ∠F, then △ABC ~ △DEF.