Similarity Criteria
1.0Master Similarity Criteria in Minutes
Understand the geometric principles used to analyze scaling and proportions. Learn how to determine if two triangles share the exact same shape regardless of size, explore the three major mathematical criteria, and discover how to apply proportional ratios to solve real-world board exam problems.
2.0Learning Outcomes
After completing this lesson, you will be able to:
- Define geometric similarity and state the core conditions for similar polygons.
- Identify and mathematically apply the AA, SAS, and SSS similarity criteria.
- Express similarity relationships using the correct symbolic notation ($\sim$).
- Calculate unknown side lengths using side-proportionality equations.
- Relate the ratio of the areas of similar triangles to the ratio of their corresponding sides.
Introduction to 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.
3.0What 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.
4.0Similar 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:
- AA (Angle-Angle) Similarity Criterion
- SAS (Side-Angle-Side) Similarity Criterion
- SSS (Side-Side-Side) Similarity Criterion
Let us understand each criterion in detail.
1. AA Similarity Criterion
AA stands for Angle-Angle Similarity.
Statement: If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar.
Why 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.
Example 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.
Statement: 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.
Condition AB/DE = AC/DF
and
∠A = ∠D
Then:
△ABC ∼ △DEF
Example 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.
Statement: If the corresponding sides of two triangles are proportional, then the triangles are similar.
Condition AB/DE = BC/EF = AC/DF
Then:
△ABC ∼ △DEF
Example 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:
1. Corresponding Angles are Equal
If: △ABC ∼ △DEF
Then:
∠A = ∠D
∠B = ∠E
∠C = ∠F
2. Corresponding Sides are Proportional
AB/DE = BC/EF = AC/DF
3. 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.
1. Measuring Heights: Surveyors use similar triangles to find the height of towers, buildings, and trees.
2. Map Scaling: Maps use similarity to represent large areas accurately on paper.
3. Architecture: Architects create scaled models of buildings using similar figures.
4. Engineering: Engineers use similarity principles in design and construction.
5. Photography: Camera lenses and image projections rely on similar triangles.
Solved Examples
Example 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.
Example 2
In two triangles:
∠A = ∠D = 70°
∠B = ∠E = 50°
Solution:
Two corresponding angles are equal.
Therefore:
Triangles are similar by AA criterion.
Example 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.
Example 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
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6.0Supporting Study Materials
This study material, including CBSE Notes and NCERT Solutions for the Chapter "Triangles" focusing on Similarity Criteria, is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It delivers precise geometric proofs, numeric scale breakdowns, and high-yield question frameworks to guarantee perfect execution in your exams.
7.030-Second Quick Revision: Similarity Criterion
- Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.
- AA (Angle-Angle) Criterion: Two angles of one triangle are equal to two angles of another triangle.
- SSS (Side-Side-Side) Criterion: Corresponding sides of two triangles are proportional.
- SAS (Side-Angle-Side) Criterion: Two sides are proportional and the included angle is equal.
- Similar triangles have the same shape but may differ in size.
- Ratio of corresponding sides remains constant.
- Corresponding angles of similar triangles are equal.
- Areas of similar triangles are proportional to the squares of corresponding sides.
- Similarity is widely used in height and distance problems.
- Symbol for similarity: ∼
8.0Previous Year's Question on Triangles
Question: In ΔABC and ΔDEF, ∠A = ∠D, ∠B = ∠E and ∠C = ∠F. State whether the triangles are similar and give the criterion used.
Solution: Since all corresponding angles are equal,
∠A = ∠D
∠B = ∠E
∠C = ∠F
Therefore, ΔABC ∼ ΔDEF by the AA Similarity Criterion.
Question: The sides of two triangles are in the ratio 3 : 4 : 5 and 6 : 8 : 10. Are the triangles similar?
Solution:
3/6 = 4/8 = 5/10 = 1/2
All corresponding sides are proportional.
Therefore, the triangles are similar by the SSS Similarity Criterion.
9.0Recommended Next Topics
- Basic Proportionality Theorem (BPT)
- Areas of Similar Triangles
- Pythagoras Theorem
- Tangents to a Circle
- Constructions Based on Similarity