Triangles Class 9th Notes - Free NCERT Class 9th Maths Chapter 7 Notes

Updated on 23 Apr 2025, 11:22 AM IST

A triangle is a closed geometrical shape with three sides, three vertices, and three angles. It is a polygon with the fewest sides. Triangles can be divided into two categories based on the sides and based on angles, like equilateral triangles, scalene triangles, isosceles triangles, etc. Triangles are used in many areas, such as building, art and design, locating objects, truss bridges, etc. In daily life, many things are triangular, like sandwiches, pizza, pyramids, etc.

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NCERT Class 9 Maths Chapter 7 Triangles: Notes
Class 9 Chapter Wise Notes
NCERT Solutions for Class 9
NCERT Exemplar Solutions for Class 9
NCERT Books and Syllabus

Triangles

These notes cover the basic definition of the triangle, triangle congruence, SAS, ASA, SSS and RHS Congruence Rule, and triangle Properties, triangle Inequalities. CBSE Class 9 chapter includes examples for the given topics as required. Students must practice all the questions and examples to ace the topic. NCERT class 9th maths notes contain all the chapters as per the latest syllabus. Students can download NCERT notes according to their standard and subjects.

NCERT Class 9 Maths Chapter 7 Triangles: Notes

Triangle

The word triangle comes from the Latin word triangulus, and the meaning of triangulus is three-cornered or having three angles. Therefore, the triangle is a closed figure that has three sides.

Congruent Triangles

If in a pair of triangles all three sides as well as three angles are equal, then this is called the congruence of triangles, and it is said Corresponding parts of the congruent triangles (CPCT), and it is denoted by the symbol $≅$. To prove the congruence of triangles, there are some criteria, and these criteria are discussed below.

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Criteria for Congruence of Triangles

Triangle congruence is one of the important concepts not only in mathematics but also in real life. For example, congruency is used to create identical objects. For defining the congruency of triangles, there are some criteria, and these are as follows.
1. SAS
2. ASA
3. AAS
4. SSS
5. RHS

SAS Criteria for Congruency

If the two sides of the pair of triangles and the angle that is included by these two sides are equal to the corresponding sides and angle that includes these two sides, then this is called Side-Angle-Side congruence (SAS), as shown in the figure.

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ASA Criteria for Congruency

If two angles of two triangles and the side that includes these two angles are equal to the corresponding two angles and the side that includes these angles is equal is called the Angle-Side-Angle congruence (ASA), as shown in the figure.

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AAS Criteria for Congruency

If in two triangles, two angles and one side are equal to the two angles and one side of another triangle, then this is called the Angle-Angle-Side Congruence (AAS), as shown in the figure.

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SSS Criteria for Congruency

If three sides of the two triangles are equal, then the two triangles then this is called the Side-Side-Side congruency(SSS), as shown in the figure.

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RHS Criteria for Congruency

If in two right-angle triangles the right angle, hypotenuse and the other side are equal, then this is called the Right-Hypotenuse-Side congruency (RHS) as shown in the figure.

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Important Note about SSA and AAA Congruency Rules

1. SSA is not a valid rule as it does not include the angle between the pair of sides
2, AAA is not a valid rule, as all three angles of both triangles may be equal, but the lengths of the sides are different.
As shown in the figure,

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Properties of a Triangle

1. Angles opposite to the equal sides of the triangle are equal in an isosceles triangle.
2. If any two sides are unequal in a triangle, then the angle opposite to the longer side of the triangle will be greater than the angle opposite to the shorter side.
3. The sum of any two sides of the triangle is greater than the size of the third side.

Class 9 Chapter Wise Notes

Students must download the notes below for each chapter to ace the topics.

NCERT Notes for Class 9 Maths Chapter 1 - Number System

NCERT Notes for Class 9 Maths Chapter 2 - Polynomials

NCERT Notes for Class 9 Maths Chapter 3 - Coordinate Geometry

NCERT Notes for Class 9 Maths Chapter 4 - Linear Equations in Two Variables

NCERT Notes for Class 9 Maths Chapter 5 - Introduction to Euclid’s Geometry

NCERT Notes for Class 9 Maths Chapter 6 - Lines And Angles

NCERT Notes for Class 9 Maths Chapter 7 - Triangles

NCERT Notes for Class 9 Maths Chapter 8 - Quadrilaterals

NCERT Notes for Class 9 Maths Chapter 9 - Circles

NCERT Notes Class 9 Maths Chapter 10 - Heron's Formula

NCERT Notes for Class 9 Maths Chapter 11 - Surface Areas and Volumes

NCERT Notes for Class 9 Maths Chapter 12 - Statistics

NCERT Solutions for Class 9

Students must check the NCERT solutions for Class 10 Maths and Science given below:

NCERT Exemplar Solutions for Class 9

Students must check the NCERT exemplar solutions for Class 10 Maths and Science given below:

NCERT Books and Syllabus

To learn about the NCERT books and syllabus, read the following articles and get a direct link to download them.

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Popular Questions

A block of mass 0.50 kg is moving with a speed of 2.00 ms^-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

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A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up ? Fat supplies 3.8×10⁷J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take g = 9.8 ms⁻² :

Option 1)

2.45×10⁻³ kg

Option 2)

6.45×10⁻³ kg

Option 3)

9.89×10⁻³ kg

Option 4)

12.89×10⁻³ kg

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An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

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A particle is projected at 60⁰ to the horizontal with a kinetic energy $K$ . The kinetic energy at the highest point

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

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In the reaction,

$2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

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How many moles of magnesium phosphate, $Mg_{3}(PO_{4})_{2}$ will contain 0.25 mole of oxygen atoms?

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

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If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

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With increase of temperature, which of these changes?

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

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Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol^-1) is

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

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A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t²) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m² , the number of rotations made by the pulley before its direction of motion if reversed, is