Similarity of Triangles

Triangles having the same shape but different sizes are known as similar triangles. Two congruent triangles are always similar but similar triangles need not be congruent. Two geometrical figures having exactly the same shape and size are said to be congruent figures. We have learned about congruent figures earlier too. Congruent figures are alike in every respect. Two triangles are said to be congruent if the sides and angles of one triangle are exactly equal to the corresponding sides and angles of the other triangle. In this article, we will be studying the similarity of triangles.

Let us study the similarity of triangles, properties of similar triangles, similarity triangles examples, similarity triangle theorem, and similarity triangle theorem proof.

Definition of Similar Triangles

Similar triangles are the triangles that look similar to each other but they might not be exactly the same in their sizes, two objects (or triangles in this case) can be said to be similar in geometry only if they have the same shape but might vary in size. 

Two triangles are said to be similar, if  

(i) their corresponding angles are equal and

(ii)their corresponding sides are proportional.

i.e Two triangles ABC and DEF are similar if 

1. ∠ A = ∠ D;  ∠ B = ∠ E;  ∠ C = ∠ F;  and

2. \[\frac{AB}{DE} = \frac{BC}{EF}  = \frac{AC}{DF}\]

The symbol for showing the similarity of triangles is ‘∼’. We can write similar triangles as 

△ABC ∼ △DEF

Properties of Similar Triangles

Similar triangles have the following properties:

• Similar triangles have the same shape but are not of the same size.

• Each corresponding pair of angles of the two similar triangles is equal.

• The ratio of any pair of corresponding sides of similar triangles is the same.

• The ratio of the area of a pair of triangles is equal to the ratio of the square of the measurements of any pair of the corresponding sides.

Similarity Triangle Theorems

Two triangles are said to be similar if any of the similarity triangle theorems are proved. These theorems are basically like a criterion for a pair of triangles to pass in order to be considered as a pair of similar triangles. These theorems let the student quickly identify whether a pair of triangles is similar in terms of geometry.

• AAA Similarity Criterion: If two triangles are equiangular, then they are similar. By equiangular, it means that the measurements of the corresponding angles of the two triangles are equal. It is also called the Angle Angle Similarity Theorem.

• SAS Similarity Criterion: If in two triangles, two pairs of corresponding sides are proportional and the included angles are equal then the two triangles are similar.  It is also called the Side Angle Side Similarity Theorem.

• SSS Similarity Criterion: If the corresponding sides of two triangles are proportional, then they are similar. It is also called the Side Side Side Similarity Theorem.

AA Similarity (Angle-Angle-Side) Criterion

The AA Similarity Criterion states that if two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

This is also sometimes called the AAA rule because equality of two corresponding pairs of angles would imply that the third corresponding pair of angles are also equal.

In the above figure,

  ∠ A = ∠ D 

  ∠ C = ∠ F 

Then  △ABC ∼ △DEF …….by AA rule

SAS Similarity (Side-Angle-Side) Criterion

SAS Similarity Criterion states that If two sides of one triangle are in proportion with the two sides of the other triangle and also one included angle between the sides is equal to the included angle of another triangle then the two triangles are similar.

In the above figure,  

\[\frac{LM}{QR} = \frac{LN}{QS}\]

And the angle between the sides are equal

i.e ∠ L = ∠Q, therefore, △MLN ∼ △RQS …….by SAS rule

SSS Similarity (Side-Side-Side) Criterion

SSS Similarity Criterion states that if the sides of one triangle are proportional or in the same ratio to the sides of another triangle then the two triangles are similar.

In the above figure,

\[\frac{AB}{DE} = \frac{BC}{EF}  = \frac{AC}{DF}\]

therefore △ABC ∼ △DEF …….by SSS rule

NOTE: It must be noted the similarity of two triangles should also be expressed symbolically, using correct correspondence of their vertices. For example, for the △ABC and △DEF, we cannot write Δ ABC ∼ Δ EDF or Δ ABC ∼ Δ FED. But, we can write Δ BAC ∼ Δ EDF.

Basic Proportionality Theorem (Thales Theorem)

The Basic Proportionality Theorem was stated by Thales, a Greek mathematician. Hence it is also known as Thales Theorem. It is abbreviated as BPT.

The Basic Proportionality Theorem States that:

If a line is parallel to a side of a triangle that intersects the other sides into two distinct points, then the line divides those sides in proportion. 

In the above figure, if we consider DE is parallel to BC, then according to the theorem,

\[\frac{AD}{DB} = \frac{AE}{EC}\]

Given: In  ΔABC, DE is parallel to BC

DE intersects sides AB and AC in points D and E respectively.

To prove \[\frac{AD}{DB} = \frac{AE}{EC}\]  

Construction:  Draw EG ⟂ AB and DF⟂ AC and join the segments BE and CD. 

Proof:

Since EG ⟂ AB. EG is the height of the ADE and DBE.

Now, we have

Area of Triangle =\[ \frac{1}{2} \times base \times height \]

Therefore, Area(ΔADE) = \[ \frac{1}{2} (AD \times EG) \]

and Area(ΔDBE) = \[ \frac{1}{2} (DB \times EG) \]

Now taking the proportions

\[\frac{Area of \Delta  ADE}{Area of \Delta DBE } = \frac{\frac{1}{2}(AD \times EG)}{\frac{1}{2}(DB \times EG)} = \frac{AD}{DB}....(1)\]

similarly, we have

\[\frac{Area of \Delta  ADE}{Area of \Delta DEC } = \frac{\frac{1}{2}(AE \times DF)}{\frac{1}{2}(EC \times DF)} = \frac{AE}{EC}....(2)\]

But the ΔDBE and ΔDEC are on the same base DE and between the same parallels DE and BC

Therefore,  

Area(ΔDBE) = Area(ΔDEC)

Taking reciprocal on both sides

\[\frac{1}{Area of \Delta DBE} = \frac{1}{ Area of \Delta DEC}\]

Multiplying both sides by Area(ΔADE)

\[\frac{Area of \Delta ADE}{Area of \Delta  DBE} = \frac{Area of \Delta ADE}{Area of \Delta  DEC}\]

Using equation 1 and 2 we get

\[ \frac{AD}{DB} = \frac{AE}{EC}\]

Hence proved

Converse of Basic Proportionality Theorem

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side i.e if \[ \frac{AD}{DB} = \frac{AE}{EC}\], then DE is parallel to BC

We will use this similarity triangle theorem-proof to solve similarity triangles examples.

Similarity of Triangles Solved Examples

Example 1: Given below are the two triangles, prove that the two triangles are similar.

Solution:

As both the triangles have two angles equal i.e 170 and 1140

So by the AA similarity theorem, we can say that the two triangles are similar.

Example 2: Prove that ABC and DEF are similar.

Solution:

In ΔABC and ΔXYZ

∠ A = ∠ X = 750

And \[ \frac{AB}{XY} = \frac{15}{10} = \frac{3}{2}\]

\[ \frac{AC}{XZ} = \frac{21}{14} = \frac{3}{2}\]

Therefore, \[ \frac{AB}{XY} = \frac{AC}{XZ}\] 

Hence by SAS Similarity, we get ΔABC ∼ ΔXYZ

Try some more similarity triangles examples on your own.

Quiz Time

1. Given that the two triangles are similar. Find the value of s.

2. Prove that the two triangles are similar.

Conclusion

This is all about the theorems, explanation, and solved examples of the similarity of triangles. Understand the reasons why two triangles are similar to each other to solve the problems easily.