Similarity of Triangles

Similarity Triangle Theorem

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

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 

(i) ∠ A = ∠ D;  ∠ B = ∠ E;  ∠ C = ∠ F;  and

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

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

△ABC ∼ △DEF

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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.


Similarity Triangle Theorems

Two triangles are said to be similar if any of the similarity triangle theorems

are proved.

  • AAA Similarity Criterion: If two triangles are equiangular, then they are similar.

  • 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.

  • SSS Similarity Criterion: If the corresponding sides of two triangles are proportional, then they are similar


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.

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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.

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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.

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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)

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


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. 

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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= \[\frac{1}{2}\] × base × height

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

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

Now taking the proportions


\[\frac{Area of \triangle{ADE}}{Area of \triangle{DBE}}\]= ½ (AD x EG) / ½ (DB x EG) = AD / DB……..(1)


similarly, we have


\[\frac{Area of \triangle{ADE}}{Area of \triangle{DEC}}\]= ½ (AE x DF) / ½ (EC x DF) = 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

\[\frac{1}{Area of \triangle{DBE}}\] = \[\frac{1}{Area of \triangle{DBE}}\]


Multiplying both sides by Area(ΔADE)


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

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 these similarity triangle theorem-proof to solve similarity triangles examples.


Solved Examples

Similarity Triangles Examples

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

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Solution:

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

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


Example 2: Prove that ABC and DEF are similar.

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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.

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  1. Prove that the two triangles are similar.

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FAQ (Frequently Asked Questions)

1. What are Congruent Triangles?

Answer: Congruent means exactly the same replica of one another. Congruent means equal in all respects or figures whose shapes and sizes are both the same. Two triangles are congruent to each other if one is superimposed on the other triangle it exactly covers one another.

If all three sides and all the three angles of one triangle are equal to corresponding sides and angles of another triangle, then the two triangles are congruent to each other. Congruence of two triangles is represented by the symbol ≅.