Similarity Criteria

The concept of Similarity Criteria is one of the most important topics in geometry. It helps us determine whether two triangles have the same shape, even if their sizes are different. Similar triangles are widely used in mathematics, architecture, engineering, surveying, and map-making.

Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are proportional. Instead of comparing every side and angle, mathematicians use specific rules known as Similarity Criteria to prove that two triangles are similar.

For Class 10 students and competitive exam aspirants, understanding similarity criteria is essential because it forms the basis for solving many geometry problems involving lengths, heights, distances, and proportional relationships.

1.0What is Similarity?

Two figures are called similar when they have the same shape but may differ in size.

For triangles:

• Corresponding angles are equal.
• Corresponding sides are proportional.

For example, if one triangle is an enlarged version of another without changing its shape, the two triangles are similar.

2.0Similar Triangles

Consider two triangles:

△ABC and △DEF

If:

∠A = ∠D
∠B = ∠E
∠C = ∠F

and

AB/DE = BC/EF = AC/DF

then:

△ABC ∼ △DEF

The symbol "∼" means "is similar to."

Similarity Criteria of Triangles

There are three main Similarity Criteria used to prove that two triangles are similar:

1. AA (Angle-Angle) Similarity Criterion
2. SAS (Side-Angle-Side) Similarity Criterion
3. SSS (Side-Side-Side) Similarity Criterion

Let us understand each criterion in detail.

1. AA Similarity Criterion

AA stands for Angle-Angle Similarity.

3.0Statement

If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar.

4.0Why Does It Work?

The sum of angles in a triangle is always 180°.

If two angles are equal, the third angle automatically becomes equal.

Therefore, all corresponding angles become equal, making the triangles similar.

5.0Example

In △ABC and △DEF:

∠A = ∠D = 60°

∠B = ∠E = 50°

Then:

∠C = ∠F = 70°

Therefore:

△ABC ∼ △DEF

This is the simplest and most commonly used similarity criterion.

2. SAS Similarity Criterion

SAS stands for Side-Angle-Side Similarity.

6.0Statement

If one angle of a triangle is equal to the corresponding angle of another triangle and the sides including those angles are proportional, then the triangles are similar.

7.0Condition

AB/DE = AC/DF

and

∠A = ∠D

Then:

△ABC ∼ △DEF

8.0Example

Given:

AB = 6 cm

AC = 9 cm

DE = 4 cm

DF = 6 cm

Check proportionality:

AB/DE = 6/4 = 3/2

AC/DF = 9/6 = 3/2

The ratios are equal and the included angles are equal.

Hence:

△ABC ∼ △DEF

3. SSS Similarity Criterion

SSS stands for Side-Side-Side Similarity.

9.0Statement

If the corresponding sides of two triangles are proportional, then the triangles are similar.

10.0Condition

AB/DE = BC/EF = AC/DF

Then:

△ABC ∼ △DEF

11.0Example

Triangle ABC:

AB = 6 cm

BC = 8 cm

AC = 10 cm

Triangle DEF:

DE = 3 cm

EF = 4 cm

DF = 5 cm

Checking ratios:

AB/DE = 6/3 = 2

BC/EF = 8/4 = 2

AC/DF = 10/5 = 2

All ratios are equal.

Therefore:

△ABC ∼ △DEF

Summary of Similarity Criteria

Properties of Similar Triangles

When two triangles are similar:

12.01. Corresponding Angles are Equal

If:

△ABC ∼ △DEF

Then:

∠A = ∠D

∠B = ∠E

∠C = ∠F

13.02. Corresponding Sides are Proportional

AB/DE = BC/EF = AC/DF

14.03. Ratio of Areas

If corresponding sides are in the ratio m:n, then areas are in the ratio m²:n².

Example

Side ratio = 2:3

Area ratio = 4:9

Applications of Similarity Criteria

Similarity criteria are used in many practical situations.

15.01. Measuring Heights

Surveyors use similar triangles to find the height of towers, buildings, and trees.

16.02. Map Scaling

Maps use similarity to represent large areas accurately on paper.

17.03. Architecture

Architects create scaled models of buildings using similar figures.

18.04. Engineering

Engineers use similarity principles in design and construction.

19.05. Photography

Camera lenses and image projections rely on similar triangles.

Solved Examples

20.0Example 1

Determine whether the triangles are similar.

Triangle 1:

Sides = 3 cm, 4 cm, 5 cm

Triangle 2:

Sides = 6 cm, 8 cm, 10 cm

Solution:

3/6 = 4/8 = 5/10 = 1/2

All ratios are equal.

Therefore:

Triangles are similar by SSS criterion.

21.0Example 2

In two triangles:

∠A = ∠D = 70°

∠B = ∠E = 50°

Solution:

Two corresponding angles are equal.

Therefore:

Triangles are similar by AA criterion.

22.0Example 3

AB = 8 cm, AC = 12 cm

DE = 4 cm, DF = 6 cm

and ∠A = ∠D

Check:

AB/DE = 8/4 = 2

AC/DF = 12/6 = 2

Sides are proportional and included angles are equal.

Therefore:

Triangles are similar by SAS criterion.

23.0Example 4

The sides of two triangles are:

5 cm, 7 cm, 9 cm

10 cm, 14 cm, 18 cm

Checking:

5/10 = 7/14 = 9/18 = 1/2

Therefore:

Triangles are similar by SSS criterion.

Example 5

A pole casts a shadow of 4 m. A nearby tower casts a shadow of 20 m. If the pole is 3 m high, find the tower's height.

Using similarity:

Height of tower / Height of pole = Shadow of tower / Shadow of pole

Tower height / 3 = 20 / 4

Tower height = 15 m

Answer: 15 m

24.0Common Mistakes Students Make

While solving similarity problems, students often make these errors:

• Comparing sides in the wrong order.
• Forgetting to check proportionality.
• Using congruence rules instead of similarity criteria.
• Ignoring corresponding angles.
• Making calculation errors in ratios.
• Mixing up AA, SAS, and SSS criteria.

To avoid these mistakes, always identify corresponding sides and angles carefully.