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Triangles Class 10th Notes - Free NCERT Class 10 Maths Chapter 6 Notes - Download PDF

Triangles Class 10th Notes - Free NCERT Class 10 Maths Chapter 6 Notes - Download PDF

Edited By safeer | Updated on Apr 08, 2025 11:21 AM IST

Geometry relies on triangles as one of its fundamental elements because they serve as foundations for multiple mathematical definitions, along with practical industrial uses. A triangle represents a fundamental closed polygon having three sides together with three angles, while also having three vertices, which have crucial significance in mathematics. This section discusses triangle classifications along with the Pythagoras theorem and the Basic Proportionality theorem, and describes the applications of triangle similarity in direct measurement calculations.

This Story also Contains
  1. NCERT notes Class 10 Maths Chapter 6 Triangles
  2. Triangles:
  3. Similarity of Polygons:
  4. Difference Between Congruent and Similar Figures:
  5. Conditions for Similarity of Triangles:
  6. Theorems on Similarity of Triangles:
  7. Area of Similar Triangles:
  8. Class 10 Chapter Wise Notes
Triangles Class 10th Notes - Free NCERT Class 10 Maths Chapter 6 Notes - Download PDF
Triangles Class 10th Notes - Free NCERT Class 10 Maths Chapter 6 Notes - Download PDF

Analysis of triangles remains vital for solving advanced mathematical problems and enables applications throughout construction and physics, and navigation disciplines. Students must practice all the topics of Linear Equations in Two Variables and their examples from the NCERT Exemplar Solutions for Class 10 Maths Chapter 6 Triangles. Students must practice questions and check solutions from the NCERT Solutions for Class 10 Maths Chapter 6 Triangles.

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NCERT notes Class 10 Maths Chapter 6 Triangles

Triangles:

A triangle exists as a polygon closed by three sides, three angles, and three vertices. Every triangle contains interior angle measures that total up to 180 degrees, and exterior angles total up to 360 degrees.

Types of Triangles:

Triangles can be classified based on sides and angles:

Based on Sides:

There are different triangle types which depend on the length measurements of their sides. The classification system helps both understand properties and solve related problems more easily. Based on the sides triangle is divided into three types:

  • Scalene Triangle: No sides are equal.

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  • Isosceles Triangle: Two sides are equal.
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  • Equilateral Triangle: All three sides are equal.

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Based on Angles:

Relating to angles forms an additional way to categorize triangles. The classification of triangle shapes allows a better understanding of angle characteristics while solving geometric problems. On the basis of angles triangle is divided into 3 types:

  • Acute Triangle: All angles are less than 90°

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  • Right Triangle: One angle is exactly 90°

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  • Obtuse Triangle: One angle is greater than 90°

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Similarity of Polygons:

In geometry, two shapes qualify as similar if they share equivalent shapes regardless of dimensional differences. In the case of polygons, two polygons are said to be similar if:

  • Their corresponding angles are equal.
  • The ratios between corresponding triangle sides remain equal.

Therefore, this means that if two triangles are given ∆ABC and ∆DEF, they are said to be similar if:

1) ∠A =∠D, ∠B =∠E and ∠C =∠F

As like:-

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OR

2) ABDE=BCEF=ACDF

As like:-

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Difference Between Congruent and Similar Figures:

  • Congruent Figures:

Two or more triangles are congruent if they have the same size and shape. Congruence is denoted by the symbol "≅" and read as "is congruent to". The shapes perfectly fit together when placed on top of each other. Congruent triangles can be flipped, rotated and mirrored and will still be congruent. Example: Two copies of the same triangle.

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△ABC ≅ △DEF

  • Similar Figures:

Two similar figures have the same shape but not necessarily the same size. One figure may exist at a different scale than the other while retaining the same basic dimensions. Similarity is denoted by the sign “~”. Example: A small triangle inside a bigger triangle with the same angles but different side lengths.

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△ABC ~ △DEF

Observation:

From the above, we can say that:

  • All congruent figures are similar.
  • Similar figures need not be congruent.

Conditions for Similarity of Triangles:

To check whether two triangles are similar, there are three conditions:

1. AA (Angle-Angle) Similarity Criterion:

If two triangles have their corresponding two angles equal, then the triangles are said to be similar triangles. This happens because when two angles match, the third angle will automatically match. After all, every triangle features a 180° angle total.

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2. SSS (Side-Side-Side) Similarity Criterion:

Two triangles are similar when their corresponding sides maintain the same ratio because this proportionality results in equal corresponding angles. Example: If in one triangle, the sides are 3 cm, 6 cm, and 9 cm, and in another triangle, the sides are 6 cm, 12 cm, and 18 cm, then the ratio is the same (1:2), so they are similar.

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3. SAS (Side-Angle-Side) Similarity Criterion:

Two triangles are similar when they share equal angles between corresponding sides, which also stand in an equal proportional relation. Example: If one triangle has sides 5 cm and 10 cm with an included angle of 50°, and another triangle has sides 10 cm and 20 cm with an included angle of 50°, they are similar.


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Theorems on Similarity of Triangles:

Basic Proportionality Theorem (Thales’ Theorem)

The Basic Proportionality Theorem (BPT), also known as Thales’ Theorem, states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Mathematical Representation:

Consider ΔABC, where DE is a line parallel to BC, and it cuts AB at D and AC at E. Then, according to the theorem:

ADDB=AEEC

Proof:

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  1. Since DE is parallel to BC, ∠ADE = ∠ABC and ∠DEA = ∠BCA (corresponding angles).
  2. By AA similarity, ΔADE is similar to ΔABC.
  3. As a result of similarity,
    ADDB=AEEC

Pythagoras Theorem:

As per Pythagoras' Theorem, “In a right-angled triangle, the sum of squares of two sides of a right triangle is equal to the square of the hypotenuse of the triangle.”

This means:

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(Hypotenuse)2 = (Base)2 + (Perpendicular)2

Area of Similar Triangles:

When the two similar triangles are given, then the square of the ratio of their corresponding sides will be equal to the ratio of their area.

Areas of similar Triangles

If ∆ABC ~ ∆PQR, then

ar(ABC)ar(PQR)=ABPQ2=BCQR2=ACPR2

Class 10 Chapter Wise Notes

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


NCERT Solutions Subject Wise

Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.

Subject-wise NCERT Exemplar Solutions

Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.

NCERT Books and Syllabus

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

Frequently Asked Questions (FAQs)

1. What is the main topic of Chapter 6, Triangles in Class 10 Maths NCERT?

The sixth chapter of NCERT Class 10 Maths textbook focuses on demonstrating the concepts of similar and congruent triangles. This chapter offers multiple theorems about triangle similarity which contain the Pythagoras theorem and the Basic Proportionality Theorem (Thales Theorem) and advanced methods to establish triangular similarity.

2. What are the key properties of triangles in Class 10 Maths?
  • The total measure of interior angles in a triangle always amounts to 180 degrees.
  • In a triangle, the exterior angle contains the total measurement of its two opposite interior angles.
  • For a triangle, all combinations of two sides should measure higher than the remaining third side (Triangle Inequality Theorem).
  • Right-angled triangles function under the principles of Pythagoras' theorem.
  • Similar triangles share equal corresponding angles in addition to having proportions that match between corresponding sides.
3. What is the Pythagoras theorem in Class 10 Chapter 6?

The Pythagoras theorem states that in a right-angled triangle:

(Hypotenuse)2 = (Base)2 + (Height)2

To solve the problems related to right-angled triangles, this theorem is used.

4. How do you prove the similarity of triangles in Class 10?

Two triangles demonstrate similarity when all pairs of angles match in size and the lengths of their sides maintain identical ratios. The proof of similarity requires the following:

  • AA (Angle-Angle) Similarity Criterion: The two triangles share similarity when they have matching angles which equal those in the other triangle.
  • SSS (Side-Side-Side) Similarity Criterion: Two triangles are similar when their corresponding sides have equivalent ratios.
  • SAS (Side-Angle-Side) Similarity Criterion: A pair of triangles is similar if two matching sides have proportional length ratios and an angle between them are equal.
5. What are the different types of triangles based on sides and angles?
  • Based on Sides:
    Scalene Triangle: No sides are equal.
    Isosceles Triangle: Two sides are equal.
    Equilateral Triangle: All three sides are equal.
  • Based on Angles:
    Acute Triangle: All angles are less than 90°.
    Right Triangle: One angle is exactly 90°.
    Obtuse Triangle: One angle is more than 90°.

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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

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 3.8×107 J 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

 

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

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

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 .

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

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.

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.

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

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

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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