Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Geometry → Similarity ↓


Criteria for Similarity


In geometry, similarity refers to a situation where two figures have the same shape but may differ in size. When two objects share the same shape, their angles are the same and the sides of one are proportional to the sides of the other. Understanding similarity is essential in solving various problems related to shapes and sizes. Similar polygons are the main focus, especially triangles. Let's understand this concept in more depth.

Understanding equality

Two geometric figures are similar if they have the same shape. This does not mean that they have the same size. Let's start with the primary criterion that determines whether two triangles are similar:

  • Angle-Angle (AA)
  • Side-Angle-Side (SAS)
  • Side-Side-Side (SSS)

Angle-Angle (AA) criterion

According to the angle-angle (AA) criterion, if two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. It is based on the fact that the sum of the interior angles in a triangle is always 180 degrees, so if two angles are known, the third angle is implicitly defined.

A B C

This triangle named ABC, where ∠A = ∠D and ∠B = ∠E, is similar to another triangle DEF under the AA criterion.

Side-Angle-Side (SAS) criterion

The Side-Angle-Side (SAS) criterion for similarity states that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in proportion, then the triangles are similar. This criterion involves two pairs of sides and the angle between them.

    Triangles ABC and DEF:
    AB / DE = BC / EF = AC / DF

Provided that the above ratios satisfy ∠A = ∠D, then triangle ABC is similar to triangle DEF according to the SAS criterion.

Side-Side-Side (SSS) criterion

According to the Side-Side-Side (SSS) criterion, two triangles are similar if the corresponding sides of the two triangles are proportional. This means that the ratio of the lengths of their corresponding sides is equal.

    Triangles ABC and DEF:
    AB / DE = BC / EF = AC / DF

If the sides of two triangles satisfy this condition, then they are similar according to the SSS criterion.

Exploring example problems

Example 1: Proving triangle similarity using the AA criterion

A triangle whose angles are ∠P = 30°, ∠Q = 60°, and another triangle whose angles are ∠A = 30°, ∠B = 60° are given. Prove that these triangles are similar.

Solution:

Since ∠P = 30° = ∠A and ∠Q = 60° = ∠B, by AA criterion, triangle PQR is similar to triangle ABC.

Example 2: Proving triangle similarity using SAS criteria

Consider two triangles: In triangle XYZ, ∠XY = 3 cm, ∠X = 45°, ∠YZ = 4 cm. In triangle DEF, DE = 6 cm, ∠D = 45°, EF = 8 cm. Prove that the triangles are similar.

Solution:

  • Check the proportionality of the sides: XY / DE = 3 / 6 = 1/2
  • XZ / DF = 4 / 8 = 1/2

Since ratios are equal and ∠X = ∠D = 45°, triangle XYZ is similar to triangle DEF by SAS criteria.

Example 3: Proving triangle similarity using the SSS criterion

Two triangles are given: triangle LMN and triangle PQR, where:

  • LM = 4 cm, MN = 5 cm, LN = 6 cm
  • PQ = 8 cm, QR = 10 cm, PR = 12 cm

Show that these triangles are similar.

Solution:

  • Check the ratio: LM / PQ = 4 / 8 = 1/2
  • MN / QR = 5 / 10 = 1/2
  • Ln / Pr = 6 / 12 = 1/2

Since all the corresponding sides are in proportion, i.e. in the same ratio, therefore, triangle LMN is similar to triangle PQR by the SSS criterion.

Practical applications

Understanding similarity in geometry has practical applications in real life, such as map reading, architecture, and even art. From knowing how to scale objects or reduce a setup into manageable sample measurements, the concepts of similarity are universally applicable in mathematical practice as well as in alignment fields.

Conclusion

Recognizing the criteria for similarity in triangles and other geometric shapes allows us to make accurate comparisons and measurements. The AA, SAS and SSS criteria form the fundamental principles that guide us in verifying similarity. These principles are not limited to theoretical mathematics but also offer ample practicality in day-to-day life scenarios, as understanding them helps us solve various geometric problems. With a broader understanding of similarity, we can address more complex problems involving other mathematical topics.


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Criteria for Similarity

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