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.

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).

$\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}$

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). 

$\text{ To prove: }\frac{AP}{BP}=\frac{AQ}{CQ}$

Given: PQ is parallel to BC. 

Construction: Join P with C and B with Q. and construct $QN\perp AP$ and $PM\perp AQ$

Solution:

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

$\text{ So, in }△APQ\text{ and }△BPQ$

$\mathrm{ar}(△APQ)=\frac{1}{2}\times AP\times QN$

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

$△APQ\text{ and }△CPQ$

$\mathrm{ar}(△APQ)=\frac{1}{2}\times AQ\times PM$

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

Now, 

$\frac{\mathrm{ar}(APQ)}{\mathrm{ar}(BPQ)}=\frac{\frac{1}{2}\times AP\times QN}{\frac{1}{2}\times BP\times QN}=\frac{AP}{BP}$

and,

$\frac{\mathrm{ar}(APQ)}{\mathrm{ar}(CPQ)}=\frac{\frac{1}{2}\times AP\times PM}{\frac{1}{2}\times CQ\times PM}=\frac{AP}{CQ}$

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, $\frac{AP}{BP}=\frac{AP}{CQ}$

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: $\frac{AP}{BP}=\frac{AP}{CQ}$

Construction: Join P with C and B with Q. and construct $QN\perp AP$ and $PM\perp AQ$

Solution: From above we know that, 

$\frac{\mathrm{ar}(APQ)}{\mathrm{ar}(BPQ)}=\frac{\frac{1}{2}\times AP\times QN}{\frac{1}{2}\times BP\times QN}=\frac{AP}{BP}$

and,

$\frac{\mathrm{ar}(APQ)}{\mathrm{ar}(CPQ)}=\frac{\frac{1}{2}\times AP\times PM}{\frac{1}{2}\times CQ\times PM}=\frac{AP}{CQ}$

$\frac{AP}{BP}=\frac{AP}{CQ}(\text{ Given })$

So, 

$\frac{\mathrm{ar}(APQ)}{\mathrm{ar}(BPQ)}=\frac{\mathrm{ar}(APQ)}{\mathrm{ar}(CPQ)}$

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. 

$\text{ Solution: in }△POQ\text{ and }△SOR$

$\angle \mathrm{P}=\angle \mathrm{S}$

$\angle \mathrm{Q}=\angle \mathrm{R}$

$\angle \mathrm{POQ}=\angle \mathrm{SOR}\text{ (Vertically opposite angle) }$

$△POQ\sim △SOR(\mathrm{AAA})$

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 $\frac{DB}{AD}=\frac{CE}{AE}$ . Prove that $△ADE\sim △ABC.$

Solution:

$\text{ In }△ADE\text{ and }△ABC\text{. }$

$\angle \mathrm{A}=\angle \mathrm{A}(\text{ Common })$

$\frac{DB}{AD}=\frac{CE}{AE}(\text{ Given })$

Add 1 on both sides, 

$\frac{DB}{AD}+1=\frac{CE}{AE}+1$

$\frac{DB+AD}{AD}=\frac{CE+AE}{AE}$

$\frac{AB}{AD}=\frac{AC}{AE}$

$△ADE\sim △ABC\text{ (SAS) }$

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 $△DEF\sim △ABC$ .

Solution:

$\text{ In }△DEF\text{ and }△ABC$

EF = BC (Given) 

AB.DF = AC.DE (Given) 

$\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}$

$△DEF\sim △ABC(\mathbf{SSS})$

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.

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.

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.