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Similar Triangles

Last Updated : 11 Mar, 2024
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Similar triangles are triangles with the same shape but can have variable sizes. Similar triangles have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles are different from congruent triangles. Two congruent figures are always similar, but two similar figures need not be congruent.

There are three similar triangle theorems:

  • AA (or AAA) or Angle-Angle Similarity Theorem
  • SAS or Side-Angle-Side Similarity Theorem
  • SSS or Side-Side-Side Similarity Theorem

Now, let’s learn more about similar triangles and their properties with solved examples and others in detail in this article.

What are Similar Triangles?

Similar triangles are triangles that look similar to each other, but their sizes might be different. Similar objects are of the same shape but different sizes. This implies similar shapes, when magnified or demagnified, should superimpose over each other. This property of similar shapes is known as “Similarity“.

Similar Triangles Definition

Two triangles are called similar triangle if their corresponding angles are equal and the corresponding sides are in the same proportion. The corresponding angles of two similar triangles must be equal. Similar triangles can have different respective lengths of the sides of the triangle, but the ratio of lengths of corresponding sides must be the same.

When two triangles are similar it implies that:

  • All pairs of corresponding angles in the triangles are equal.
  • All pairs of corresponding sides of the triangle are proportional.

The symbol “” is used to represent the similarity between similar triangles. So, when two triangles are similar, we write it as △ABC ∼ △DEF.

Similar Triangles Examples

Various examples of the similar triangles are:

  • If we take two triangles that have sides in the ratio then they are the similar triangles.
  • The Flagpoles and their Shadows represents the similar triangles.

The triangle shown in the image below are similar triangle and we represent them as, △ABC ∼ △PQR.

Similar Triangles

Basic Proportionality Theorem (Thales Theorem)

Basic Proportionality Theorem, also known as Thales’ Theorem, is a fundamental concept in geometry that relates to the similarity of triangles. It states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. In simpler terms, if a line parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally.

Mathematically, if a line DE is drawn parallel to one side of triangle ABC, intersecting sides AB and AC at points D and E respectively, then according to the Basic Proportionality Theorem:

BD/DA = CE/EA

This theorem is a consequence of the similarity of triangles formed by the parallel line and the sides of the original triangle. Specifically, triangles ADE and ABC, as well as triangles ADC and AEB, are similar due to corresponding angles being equal. Consequently, the ratios of corresponding sides in similar triangles are equal, leading to the proportionality relationship described by the Basic Proportionality Theorem.

Basic Proportionality Theorem is widely used in geometry and trigonometry to solve various problems involving parallel lines and triangles. It serves as a foundational principle for understanding the properties of similar triangles and the relationships between their corresponding sides and angles. Additionally, it forms the basis for more advanced concepts in geometry, such as the Parallel Lines Theorem and applications in various geometric constructions and proofs.

Rules of Similar Triangles

If two triangles are similar they must meet one of the following rules,

  • Two pairs of corresponding angles are equal. (AA Rule)
  • Three pairs of corresponding sides are proportional. (SSS Rule)
  • Two pairs of corresponding sides are proportional and the corresponding angles between them are equal. (SAS Rule)

Similar Triangles Formula

In the last section, we studied two conditions using which we can verify whether the given triangles are similar or not. The conditions are when two triangles are similar; their corresponding angles are equal, or the corresponding sides are in proportion. Using either condition, we can prove △PQR and △XYZ are similar from the following set of similar triangle formulas.

Formula for Similar Triangles in Geometry

In △PQR and △XYZ if,

  1. ∠P = ∠X , ∠Q = ∠Y, ∠R = ∠Z
  2. PQ/XY = QR/YZ = RP/ZX

The above two triangles are similar, i.e., △PQR ∼ △XYZ. 

Similar Triangles Theorems (Similar Triangle Rules)

The similarity theorems help us to find whether the two triangles are similar or not. When we do not have the measure of angles or the sides of the triangles, we use the similarity theorems. 

There are three major types of similarity rules, as given below:

  • AA (or AAA) or Angle-Angle Similarity Theorem
  • SAS or Side-Angle-Side Similarity Theorem
  • SSS or Side-Side-Side Similarity Theorem

Angle-Angle (AA) or AAA Similarity Theorem

AA similarity criterion states that if any two angles in a triangle are respectively equal to any two angles of another triangle, then they must be similar triangles. AA similarity rule is easily applied when we only know the measure of the angles and have no idea about the length of the sides of the triangle. 

In the image given below, if it is known that ∠B = ∠G, and ∠C = ∠F:

Angle-Angle Similarity Criterion

And we can say that by the AA similarity criterion, △ABC and △EGF are similar or △ABC ∼ △EGF.

⇒AB/EG = BC/GF = AC/EF and ∠A = ∠E.

Side-Angle-Side or SAS Similarity Theorem

According to the SAS similarity theorem, if any two sides of the first triangle are in exact proportion to the two sides of the second triangle along with the angle formed by these two sides of the individual triangles are equal, then they must be similar triangles. This rule is generally applied when we only know the measure of two sides and the angle formed between those two sides in both triangles respectively.

In the image given below, if it is known that AB/DE = AC/DF, and ∠A = ∠D

Side-Angle-Side Similarity Criterion

And we can say that by the SAS similarity criterion, △ABC and △DEF are similar or △ABC ∼ △DEF.

Side-Side-Side or SSS Similarity Theorem

According to the SSS similarity theorem, two triangles will the similar to each other if the corresponding ratio of all the sides of the two triangles are equal. This criterion is commonly used when we only have the measure of the sides of the triangle and have less information about the angles of the triangle.

In the image given below, if it is known that PQ/ED = PR/EF = QR/DF

Side-Side-Side Similarity Criterion

And we can say that by the SSS similarity criterion, △PQR and △EDF are similar or △PQR ∼ △EDF.

Similar Triangle Properties

Similar triangles have various properties which are widely used for solving various geometrical problems. Some of the common properties of similar triangle:

  • The shape of similar triangles is fixed but their sizes may be different.
  • Corresponding angles of similar triangles are equal.
  • Corresponding sides of similar triangles are in common ratios.
  • The ratio of the area of similar triangles is equal to the square of the ratio of their corresponding side.

How to Find Similar Triangles?

Two given triangles can be proved as similar triangles using the above-given theorems. We can follow the steps given below to check if the given triangles are similar or not:

Step 1: Note down the given dimensions of the triangles (corresponding sides or corresponding angles).

Step 2: Check if these dimensions follow any of the conditions for similar triangles theorems(AA, SSS, SAS).

Step 3: The given triangles, if satisfy any of the similarity theorems, can be represented using the “∼” to denote similarity.

This can be understood better with the help of the following example:

Example: Check if △ABC and △PQR are similar triangles or not using the given data: ∠A = 65°, ∠B = 70º and ∠P = 70°, ∠R = 45°.

Using given measurement of angles, we cannot conclude if the given triangles follow the AA similarity criterion or not. Let us find the measure of the third angle and evaluate it.

We know, using the angle sum property of a triangle, ∠C in △ABC = 180° – (∠A + ∠B) = 180° – 135° = 45°

Similarly, ∠Q in △PQR = 180° – (∠P + ∠R) = 180° – 115° = 65°

Therefore, we can conclude that in △ABC and △PQR, 

∠A = ∠Q, ∠B = ∠P, and ∠C = R

△ABC ∼ △QPR

Area of Similar Triangle Theorem

Similar Triangle Area Theorem states that for two similar triangles ratio of area of the triangles is proportional to the square of the ratio of their corresponding sides. Suppose we are given two similar triangles, ΔABC and ΔPQR then

According to Similar Triangle Theorem:

(Area of ΔABC)/(Area of ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (CA/RP)2

Difference Between Similar Triangles and Congruent Triangles

Similar triangles and congruent triangles are two types of triangles that are widely used in geometry for solving various problems. Each type of triangle has different properties and the basic difference between them is discussed in the table below.

Similar Triangles

Congruent Triangles

Similar triangles are triangles that have equal corresponding angles. Congruent triangles are triangles that have equal corresponding angles and equal corresponding sides.
Similar triangles have the same shape but their sizes may or may not be the same Congruent triangles have the same size and the same area.
Similar triangles are not superimposed images of each other until magnified or demagnified. Congruent triangles are superimposed images of each other if arrange in the proper orientation.
Similar triangles are represented with the ‘~’ symbol. Congruent triangles are represented with the ‘’ symbol.
Their corresponding sides are in the ratio. Their corresponding sides are equal.

Similar Triangle Applications

Various applications of the similar triangle that we see in the real life are,

  • Shadow and Height of various objects are calculated using the concept of similar triangles.
  • Map Scaling uses the concept of the similar triangle.
  • Photographic devices uses the similar triangle properties to capture various images.
  • Model Making uses the concept of similar triangles.
  • Navigation and Trigonometry also uses the similar triangle approach to solve various problems, etc.

Related Articles:

Congruence of Triangles

Area of Triangle

Right Angle Triangle

Perimeter of Triangle

Important Notes on Similar Triangles:

  • Ratio of areas of similar triangles is equal to square of ratio of their corresponding sides.
  • All congruent triangles are similar, but all similar triangles may not necessarily be congruent.
  • This ‘~’ symbol is used to denote similar triangles.

Solved Questions on Similar Triangles

Question 1: In the given figure 1, DE || BC. If AD = 2.5 cm, DB = 3 cm, and AE = 3.75 cm. Find AC?

Example 1 - Similar Triangles

Solution:

In △ABC, DE || BC

AD/DB = AE/EC   (By Thales’ Theorem)

2.5/3 = 3.75/x, where EC = x cm

(3 × 3.75)/2.5 = 9/2 = 4.5 cm

EC = 4.5 cm

Hence, AC = (AE + EC) = 3.75 + 4.5 = 8.25 cm.

Question 2: In Figure 1 DE || BC. If AD = 1.7 cm, AB = 6.8 cm, and AC = 9 cm. Find AE?

Solution:

Let AE = x cm.

In △ABC, DE || BC

By Thales Theorem we have,

AD/AB = AE/AC

1.7/6.8 = x/9

x = (1.7×9)/6.8 = 2.25 cm

AE = 2.25 cm

Hence AE = 2.25 cm

Question 3: Prove that a line drawn through the midpoint of one side of a triangle (figure 1) parallel to another side bisects the third side.

Solution:

Given a ΔΑΒC in which D is the midpoint of AB and DE || BC, meeting AC at E.

TO PROVE AE = EC.

Proof: Since DE || BC, by Thales’ theorem, we have:

AE/AD = EC/DB =1  (AD = DB, given)

AE/EC = 1

AE = EC

Question 4: In the given Figure 2, AD/DB = AE/EC and ∠ADE = ∠ACB. Prove that ABC is an isosceles triangle.

Question 4 - Similar Triangles

Solution:

We have AD/DB = AE/EC DE || BC [by the converse of Thales’ theorem] 

∠ADE = ∠ABC (corresponding ∠s) 

But, ∠ADE = ∠ACB (given). 

Hence, ∠ABC = ∠ACB.

So, AB = AC [sides opposite to equal angles]. 

Hence, △ABC is an isosceles triangle.

Question 5: If D and E are points on the sides AB and AC respectively of △ABC  (figure 2) such that AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm, and AE = 1.8 cm, show that DE || BC.

Solution:

Given, AB = 5.6 cm, AD = 1.4 cm, AC = 7.2 cm and AE = 1.8 cm

AD/AB = 1.4/5.6 = 1/4 and AE/AC = 1.8/7.2 = 1/4

AD/AB = AE/AC

Hence, by converse of Thales Theorem, DE || BC.

Question 6: Prove that the line segment joining the midpoints of any two sides of a triangle (figure 2) is parallel to the third side.

Solution:

In △ABC in which D and E are the midpoints of AB and AC respectively. 

Since D and E are the midpoints of AB and AC respectively, we have : 

AD = DB and AE = EC.

AD/DB = AE/EC (each equal to 1)

Hence, by converse of Thales Theorem, DE || BC

Practice Questions Similar Triangles

Q1. In two similar triangle △ABC and △ADE, if DE || BC and AD = 3 cm, AB = 8 cm, and AC = 6 cm. Find AE.

Q2. In two similar triangle △ABC and △PQR, if QR || BC and PQ = 2 cm, AB = 12 cm, and AC = 9 cm. Find PR.

Q3. In two similar triangles ΔABC and ΔAPQ, the length of the sides are given as AP = 9 cm , PB = 12 cm and BC = 24 cm. Find the ratio of the areas of ΔABC and ΔAPQ.

Q4. In two similar triangles ΔABC and ΔAPQ, the length of the sides are given as AP = 3 cm , PB = 4 cm and BC = 8 cm. Find the ratio of the areas of ΔABC and ΔAPQ.

Similar Triangles – FAQs

What are Similar Triangles Class 10?

Similar triangles are the triangles which gave all the angles equal and their sides are in a common ratio. They have a similar shape but not a similar area.

What are Similar Triangles Formulas?

Similar triangle formulas are the formulas that tell us whether two triangles are similar or not. For two triangles △ABC and △XYZ, the similar triangles formula are,

  • ∠A = ∠X, ∠B = ∠Y and ∠C = ∠Z
  • AB/XY = BC/YZ = CA/ZX

Which Symbol is used for representing Similar Triangles?

Similar Triangles are represented using the ‘~’ symbol. If two triangles △ABC and △XYZ are similar we represent them as, △ABC ~ △XYZ, it is read as triangle ABC similar to triangle XYZ.

What are 3 Similar Triangle Theorems?

We can easily prove two triangles to be similar by using three triangle theorem that are,

  • AA (or AAA) or Angle-Angle Similarity Theorem
  • SAS or Side-Angle-Side Similarity Theorem
  • SSS or Side-Side-Side Similarity Theorem

What are Properties of Similar Triangles?

The important properties of the similar triangle are,

  • Similar triangles have fixed shapes but their sizes may be different.
  • Corresponding angles are equal in a similar triangle.
  • Corresponding sides are in common ratios in a similar triangle.

How to know if two Triangles are Similar?

If all the angles in a triangle are equal then we can easily say that triangles are similar.

Which Triangles are always Similar?

The triangle which is always similar is an equilateral triangle. As all the angles in the equilateral triangles are always 60 degrees any two equilateral triangles are always similar.

What is Similar Triangles Area?

The ratio of the area of two similar triangles is always equal to the ratio of squares of their sides. For two triangles △ABC and △XYZ, we can say that,

  • area △ABC / area △XYZ = (AB / XY)2

What is Similar Triangle Criteria?

Similar Triangle Criteria is the criteria in which we can declare three triangles as similar triangles and these three criteria are,

  • AAA Criteria (Angle-Angle-Criteria)
  • SAS Criteria (Side-Angle-Side Criteria)
  • SSS Criteria (Side-Side-Side Criteria)

Who is the father of similar triangles?

Euclid, the ancient Greek mathematician often referred to as the “father of geometry,” provided foundational principles for understanding similar triangles in his work “Elements.”

Are similar triangles proportional?

Yes, similar triangles are proportional. This means that the corresponding sides of similar triangles are in proportion, which implies that the ratio of corresponding sides of similar triangles remains constant.

Which triangles are always similar?

Triangles that have the same three angles are always similar. This is a fundamental property known as the Angle-Angle (AA) similarity criterion.

Are all right triangles similar?

No, not all right triangles are similar. While right triangles with the same acute angles are similar, the length of the hypotenuse and the ratio of side lengths may differ, leading to non-similarity between right triangles.

What is the ratio of two similar triangles?

The ratio of any two corresponding sides in similar triangles remains constant. This means that if you take corresponding sides of similar triangles and form a ratio, the result will always be the same, regardless of the specific side lengths chosen.



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