Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Triangles ↓


Properties of Triangles


Triangles are one of the simplest but most fundamental shapes in geometry. They are three-sided polygons with three angles. It is important to understand the properties of triangles because they form the basis for more complex geometric concepts. In this lesson, we will explore the properties and characteristics of triangles in detail.

Types of triangles

Triangles can be classified based on their sides and angles. Let's take a look at these classifications:

Types based on sides

  • Equilateral triangle: All three sides are equal, and all three angles are 60 degrees. A unique feature of equilateral triangles is that they are perfectly symmetrical. 
    Example: Sides = 5 cm, 5 cm, 5 cm; Angles = 60°, 60°, 60°
  • Isosceles triangle: Two sides are equal, and the angles opposite to these sides are also equal. 
    Example: Sides = 5 cm, 5 cm, 8 cm; Angles = 70°, 70°, 40°
  • Scalene triangle: All sides and all angles are different. 
    Example: Sides = 4 cm, 5 cm, 6 cm; Angles = 40°, 60°, 80°

Types based on angles

  • Acute triangle: All angles are less than 90 degrees. 
    Example: Angles = 50°, 60°, 70°
  • Right angle: One of its angles is exactly 90 degrees. 
    Example: Angles = 30°, 60°, 90°
  • Obtuse triangle: One of its angles is more than 90 degrees. 
    Example: Angles = 30°, 45°, 105°

Properties of triangles

Now, let's explore some fundamental properties of triangles:

Sum of interior angles

The sum of the interior angles in any triangle is always 180 degrees. This property is important when solving problems and finding missing angles.

For example:

If two angles of a triangle are 70° and 40°, the third angle is 180° - (70° + 40°) = 70°.

Exterior angle property

The exterior angle of a triangle is equal to the sum of its two opposite interior angles.

For example:

In a triangle with angles 50°, 60°, and 70°, an exterior angle adjacent to the 50° angle is 120°, because 120° = 60° + 70°.

Triangle inequality theorem

This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

  • For a triangle with sides a, b, c: 
    a + b > c, a + c > b, b + c > a

Example:

If a triangle has sides 3 cm, 4 cm, and 5 cm, it satisfies: 3 + 4 > 5, 3 + 5 > 4, and 4 + 5 > 3.

Similar triangles

Two triangles are similar if they have the same shape, which means corresponding angles are equal, and corresponding sides are in proportion.

If triangle ABC is similar to triangle DEF, then ∠A = ∠D, ∠B = ∠E, ∠C = ∠F; and AB/DE = BC/EF = CA/FD.

Congruent triangle

Two triangles are congruent if they are similar in size and shape, that is, their corresponding sides and angles are equal.

Criteria for Conformity:

  • SSS(Side-Side-Side) Criterion: If three sides of a triangle are equal to three sides of another triangle.
  • SAS (Side-Angle-Side) Criterion: If two sides and their included angle of a triangle are equal to two sides and their included angle of another triangle.
  • ASA (Angle-Side-Angle) Criterion: If two angles and the included side of a triangle are equal to two angles and the included side of another triangle.
  • AAS (Angle-Angle-Side) Criterion: If two angles and a non-included side of a triangle are equal to two angles and the corresponding non-included side of another triangle.
  • RHS (Right Angle-Hypotenuse-Side) Criterion: In right angled triangles, if the hypotenuse and a side of one triangle are equal to the hypotenuse and a side of the other triangle.

Area and perimeter

Area of triangles

The area of a triangle can be found using different formulas depending on the available information:

  • Use of base and height: 
    Area = 0.5 × base × height
  • Using Heron's formula: for sides a, b, c, semi-perimeter s = (a + b + c)/2 
    Area = √[s × (s - a) × (s - b) × (s - c)]

Example using base and height:

If a triangle has a base of 4 cm and a height of 3 cm, the area is 0.5 × 4 × 3 = 6 cm².

Perimeter of triangles

The perimeter of a triangle is the sum of its sides:

Perimeter = side1 + side2 + side3

Example:

If a triangle's sides are 5 cm, 6 cm, and 7 cm, the perimeter is 5 + 6 + 7 = 18 cm.

Special lines in triangles

  • Median: Median divides the triangle into two equal parts. It is a line drawn from a vertex to the midpoint of the opposite side.
  • Altitude: Altitude is the perpendicular line segment from a vertex to the line on the opposite side.
  • Angle bisector: An angle bisector divides an angle into two equal angles.
  • Perpendicular bisector: Perpendicular bisector is the line that divides a side into two equal parts at 90 degrees.

Conclusion

The properties of triangles are the cornerstone of many geometric concepts. By understanding the types, properties, and formulas related to triangles, one can solve a wide range of mathematical problems. The ability to identify different triangle types and apply the appropriate properties and theorems is an invaluable skill in mathematics.


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Criteria for Congruence
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Properties of Triangles

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