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Chapter 6 Triangles

Class 10

Maths

Chapter 6 Triangles

Frequently Asked Questions

Congruent triangles are identical in both shape and size but similar triangles are not.

In maths, a median cuts the side of a triangle in equal parts while altitude is perpendicular to the triangle.

The area ratio is equal to the square of the side length ratio in similar triangles.

Knowing the properties of triangles helps in geometry and real-life geometrical problems.

The notes cover basic triangle definitions, similarity of triangles, criteria for similarity (AA, SAS, SSS), proportionality relations, and applications of triangle similarity in problem solving.

Key Notes present concise summaries of theorems, formulas, and quick facts, while Detailed Notes include thorough explanations, complete proofs, and step-by-step solved examples.

Each similarity criterion is broken down with examples and hints on when to apply which test while comparing triangles in geometric problems.

They clarify how corresponding sides of similar triangles are in proportion and relate this concept to line segments connecting triangle points.

Yes, the notes include real-life and diagram-based problems involving finding missing sides, ratios, and angles using similarity criteria.

Area comparisons for triangles with equal heights or bases and their proportional relations are explained for board-oriented questions.

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CBSE Notes for Class 10 Maths Chapter 6 Triangles

CBSE Notes Class 10 Maths Chapter 6 – Triangles provide a clear understanding of important theorems and properties related to triangles. This chapter covers similarity of triangles, criteria for similarity, basic proportionality theorem (Thales’ theorem), and the Pythagoras theorem with proofs and applications. These CBSE Notes help students revise key concepts, formulas, and problem-solving methods efficiently for the board examination.

1.0Introduction to triangles

A triangle is a type of polygon with three sides. Triangles are of different types like Equilateral triangles, Isosceles triangles, and Right angle triangles.

2.0Download CBSE Notes for Class 10 Maths Chapter 6: Triangles - Free PDF!!

Get your CBSE Notes for Class 10 Maths Chapter 6: Triangles in a free PDF format for fast and effective revision. These notes include key theorems, formulas, and solved examples to help you prepare confidently for your board exams.

Class 10 Maths Chapter 6 Revision Notes:
Class 10 Maths Chapter 6 Key Notes :

3.0CBSE Class 10 Maths Chapter 6 Triangles - Revision Notes

What are similar figures?

Similar figures mean that two or more figures have the same shape but not necessarily the same size. For example: All equilateral triangles are similar but not necessarily congruent. Similarly, all squares are similar but not congruent. Here, we say that all congruent figures are similar but the similar figures need not be congruent.

Similarity in Triangles

Two triangles are similar if:

Their corresponding angles are equal.

That is Angle A = D, B = E, C = F.

Their corresponding sides are in the same ratio (or proportion).

$\frac{A B}{D E} = \frac{A C}{D F} = \frac{BC}{EF}$

Properties of Triangles

Theorem 1: 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. (Thales or basic proportionality theorem).

$To prove: \frac{A P}{BP} = \frac{A Q}{CQ}$

Given: PQ is parallel to BC.

Construction: Join P with C and B with Q. and construct $QN ⊥ A P$ and $PM ⊥ A Q$

Solution:

$Area of a triangle = \frac{1}{2} \times Base \times Height$

$So, in △ A PQ and △ BPQ$

$ar (△ A PQ) = \frac{1}{2} \times A P \times QN$

$ar (△ BPQ) = \frac{1}{2} \times BP \times QN$

$△ A PQ and △ CPQ$

$ar (△ A PQ) = \frac{1}{2} \times A Q \times PM$

$ar (△ CPQ) = \frac{1}{2} \times CQ \times PM$

Now,

$\frac{ar ( A PQ )}{ar ( BPQ )} = \frac{\frac{1}{2} \times A P \times QN}{\frac{1}{2} \times BP \times QN} = \frac{A P}{BP}$

and,

$\frac{ar ( A PQ )}{ar ( CPQ )} = \frac{\frac{1}{2} \times A P \times PM}{\frac{1}{2} \times CQ \times PM} = \frac{A P}{CQ}$

We know in maths, that the area of two triangles with the same base and between two parallel lines is equal. Hence, ar(BPQ) = ar(CPQ).

Hence, $\frac{A P}{BP} = \frac{A P}{CQ}$

Theorem 2: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. (Converse of Thales theorem)

To Prove: PQ is parallel to BC.

Given: $\frac{A P}{BP} = \frac{A P}{CQ}$

Construction: Join P with C and B with Q. and construct $QN ⊥ A P$ and $PM ⊥ A Q$

Solution: From above we know that,

$\frac{ar ( A PQ )}{ar ( BPQ )} = \frac{\frac{1}{2} \times A P \times QN}{\frac{1}{2} \times BP \times QN} = \frac{A P}{BP}$

and,

$\frac{ar ( A PQ )}{ar ( CPQ )} = \frac{\frac{1}{2} \times A P \times PM}{\frac{1}{2} \times CQ \times PM} = \frac{A P}{CQ}$

$\frac{A P}{BP} = \frac{A P}{CQ} (Given)$

So,

$\frac{ar ( A PQ )}{ar ( BPQ )} = \frac{ar ( A PQ )}{ar ( CPQ )}$

Hence,

ar(BPQ) = ar(CPQ).

If the area of two triangles between two lines with equal bases is equal to each other then the lines will be parallel. Hence, PQ is parallel to BC.

Criteria for Similarity

In Maths, if only two angles are equal then the third angle will automatically be equal.
SAS (Side-Angle-Side Theorem): Meaning if two triangles are similar then the rest of the corresponding parts will also be equal.

AAA stands for Angle-Angle-Angle meaning when all angles of two triangles are equal.

Example: In the given figure: Prove that $POQ \sim SOR$ given that PQ is parallel to RS.

$Solution: in △ POQ and △ SOR$

$∠ P = ∠ S$

$∠ Q = ∠ R$

$∠ POQ = ∠ SOR (Vertically opposite angle)$

$△ POQ \sim △ SOR (AAA)$

SAS: Stands for Side-angle-Side which means the ratio of corresponding sides is equal to an angle between these sides.

Example: In the given figure $\frac{D B}{A D} = \frac{CE}{A E}$ . Prove that $△ A D E \sim △ A BC .$

Solution:

$In △ A D E and △ A BC .$

$∠ A = ∠ A (Common)$

$\frac{D B}{A D} = \frac{CE}{A E} (Given)$

Add 1 on both sides,

$\frac{D B}{A D} + 1 = \frac{CE}{A E} + 1$

$\frac{D B + A D}{A D} = \frac{CE + A E}{A E}$

$\frac{A B}{A D} = \frac{A C}{A E}$

$△ A D E \sim △ A BC (SAS)$

SSS: Stands for Side-Side-Side which means the ratio of all the corresponding sides is equal.

Example: In the given figures BC = EF, and AB.DF = AC.DE Prove that $△ D EF \sim △ A BC$ .

Solution:

$In △ D EF and △ A BC$

EF = BC (Given)

AB.DF = AC.DE (Given)

$\frac{A B}{D E} = \frac{A C}{D F} = \frac{BC}{EF}$

$△ D EF \sim △ A BC (SSS)$

4.0CBSE Class 10 Maths Chapter 6 Triangles - Key Notes

Similar triangles

ΔABC ~ ΔPQR (~ is the sign of similarity)

Also, AB / PQ = BC / QR = CA / PR

Similar figures

Two figures are similar if these two conditions are true :

(i) Corresponding angles are congruent.

(ii) The lengths of corresponding sides are in proportion.

The proportions a / b = c / d and a : b = c : d can be read as "a is to b as c is to d".

For example, if PQRS ~ TUVW, then :

(i) ∠P ≅ ∠T, ∠Q ≅ ∠U, ∠R ≅ ∠V and ∠S ≅ ∠W

(ii) PQ / TU = QR / UV = RS / VW = SP / WT

Angle-Angle (AA) similarity postulate

If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.

Side-Angle-Side (SAS) similarity theorem

If an angle of one triangle is congruent to an angle of another triangle, and the sides including these angles are in proportion, then the triangles are similar.

If ∠A ≅ ∠D and AB / DE = AC / DF

then ΔABC ~ ΔDEF.

Side-Side-Side (SSS) similarity theorem

If all corresponding sides of two triangles are in proportion, then the triangles are similar.

If AB / DE = BC / EF = AC / DF

then ΔABC ~ ΔDEF.

Thales theorem or basic proportionality theorem

If a line is drawn parallel to one side of a triangle intersecting the other two sides, then the other two sides are divided in the same ratio.

If ST || QR, then

PS / SQ = PT / TR

Converse of basic proportionality theorem

If a line divides any two sides of a triangle proportionally, the line is parallel to the third side.

If

AD / DB = AE / EC

then DE || BC

Proportions

Two equations are equivalent if one equation can be changed into the other using algebra.

For example, a / b = c / d is equivalent to ad = bc.

Properties of proportions

All of the proportions below are equivalent to each other.

a / b = c / d ⇔ b / a = d / c ⇔ a / c = b / d ⇔ c / a = d / b

These proportions are also equivalent to the ones above. All of the proportions below are equivalent to each other.

a / (a + b) = c / (c + d)

b / (a + b) = d / (c + d)

(a + b) / a = (c + d) / c

Results on area of similar triangles

The areas of two similar triangles are proportional to the squares of their corresponding sides.

The areas of two similar triangles are proportional to the squares of their corresponding altitude.

Area of ΔABC / Area of ΔDEF = AB² / DE² = BC² / EF² = AC² / DF²

Corollary-1

The areas of two similar triangles are proportional to the squares of their corresponding altitude.

The areas of two similar triangles are proportional to the squares of their corresponding medians.

Area of ΔABC / Area of ΔDEF = AL² / DM²

Corollary-2

The areas of two similar triangles are proportional to the squares of their corresponding medians.

Area of ΔABC / Area of ΔDEF

= AP² / DQ²

Corollary-3

The areas of two similar triangles are proportional to the squares of their corresponding angle bisector segments.

Area of ΔABC / Area of ΔDEF = AX² / DY²

5.0Key Features of CBSE Maths Notes for Class 10 Chapter 6

The notes are aligned with the latest curriculum suggested by CBSE.
Step-by-step guides along with examples are given to provide a better understanding of concepts.
The language used is easy hence ideal for self-learning.

Chapter-wise CBSE Notes for Class 10 Maths :

Class 10 Maths Chapter 1 - Real Numbers Notes

Class 10 Maths Chapter 2 - Polynomials Notes

Class 10 Maths Chapter 3 - Linear Equations In Two Variables Notes

Class 10 Maths Chapter 4 - Quadratic Equations Notes

Class 10 Maths Chapter 5 - Arithmetic Progressions Notes

Class 10 Maths Chapter 6 - Triangles Notes

Class 10 Maths Chapter 7 - Coordinate Geometry Notes

Class 10 Maths Chapter 8 - Introduction To Trigonometry Notes

Class 10 Maths Chapter 9 - Some Applications of Trigonometry Notes

Class 10 Maths Chapter 10 - Circles Notes

Class 10 Maths Chapter 11 - Areas Related To Circles Notes

Class 10 Maths Chapter 12 - Surface Areas and Volumes Notes

Class 10 Maths Chapter 13 - Statistics Notes

Class 10 Maths Chapter 14 - Probability Notes

Chapter-wise NCERT Solutions Class 10 Maths:

Chapter 1 - Real Numbers

Chapter 2 - Polynomials

Chapter 3 - Linear Equations In Two Variables

Chapter 4 - Quadratic Equations

Chapter 5 - Arithmetic Progressions

Chapter 6 - Triangles

Chapter 7 - Coordinate Geometry

Chapter 8 - Introduction To Trigonometry

Chapter 9 - Some Applications of Trigonometry

Chapter 10 - Circles

Chapter 11 - Areas Related To Circles

Chapter 12 - Surface Areas and Volumes

Chapter 13 - Statistics

Chapter 14 - Probability