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Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 PDF Download

INTRODUCTION
You have learnt about triangles and some of their properties in your previous classes. In this chapter, we will study about the congruency of triangles and some more properties. We may had some idea about these properties in lower classes but here, we will study these properties in greater details.
 

CONGRUENCE OF TRIANGLES
Congruent figures : Two geometrical figures, having exactly the same shape and size are known as
congruent fig. For congruence, we use the symbol Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9.
Thus, two line segments are congruent, if they have the same length.
Two angles are congruent, if they have the same measure.

Congruent Triangles : Two triangles are congruent, if and only if one of them can be made to superpose on the other, so as to cover it exactly. 

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9


Thus, congruent triangles are exactly identical, i.e., their corresponding three sides and the three angles are
equal.
If ΔABC is congruent to ΔDEF, we write ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔDEF. This happens when AB = DE, BC = EF, AC =
DF and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9D, Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9E, Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9C = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9F.
In this case, we say that the sides corresponding to AB, BC and AC are DE, EF and DF respectively.
And, the angles corresponding to Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A, Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9C are Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9D, Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9E and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9F respectively.


Thus, the corresponding parts of two congruent triangles are equal. We show it by the abbreviation
C.P.C.T., which means corresponding parts of congruent triangles.


Congruence Relation in the Set of All Triangles : From the definition of congruence of two triangles, we obtain the following results :
(i) Every triangle is congruent to itself i.e. ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔABC.
(ii) If ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔDEF, then ΔDEF Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔABC.
(iii) if ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ΔDEF, and ΔDEF Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔPQR, then ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔPQR.


CRITERIA FOR CONGRUENCE OF TRIANGLES
In earlier classes, we have learnt some criteria for congruence of triangles. Here, in this class we will learn the truth of these either experimentally or by deductive proof.
In the previous section, we have studied that two triangles are congruent if and only if there exists a correspondence between their vertices such that the corresponding sides and the corresponding angles of two triangles are equal i.e. six equalities hold good, three of the corresponding sides and three of the corresponding angles.
In this section, we shall prove that if three properly chosen conditions out of the six conditions are satisfied, then the other three are automatically satisfied. Let us now discuss those three conditions which ensure the congruence of two angles.


I . SIDE-ANGLES-SIDE (SAS) CONGRUENCE CRITERION

AXIOM : Two triangles are congruent if two sides and included angle of one triangle are equal to the sides and the included angle of the other triangle.
In the given figure, in ΔABC and ΔDEF, we have :
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
AB = DE, AC = DF and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9D.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔDEF [By SAS-criteria]
 

REMARK : SAS Congruence rule holds but ASS or SSA rule does not hold.

II. ANGLE-SIDE-ANGLE (ASA) CONGRUENCE CRITERION

THEOREM-1 : Two triangles are congruent, if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Proof : To prove ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔDEF, we need to consider three possible situations :

Case 1. Let AB = DE

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Case 2. Let AB > DE.

Construction : take a point H on AB such that HB = DE

 

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

It can be possible only if H coincides with A. In other words, AB = DE.


Case 3. Let AB < DE.
Construction : Take a point M on DE such that ME = AB.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
By repeating the same arguments as in Case 2, we can
prove that AB = DE, and so ΔABC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔDEF.
Hence, proved.


III. ANGLE-ANGLE-SIDE (AAS) CONGRUENCE CRITERION
 

THEOREM-2 : Two triangles are congruent, if any two pairs of angles and one pair of corresponding
 sides are equal.

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Hence Proved

 

IV. SIDE-SIDE-SIDE (SSS) CONGRUENCE CRITERION

Two triangles are congruent, if the three sides of one triangle are equal to the corresponding
 three sides of the other triangle.

In the given figure, in ΔABC and ΔDEF, we have :
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
AB = DE, BC = EF and AC = DF.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔABC  Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔDEF [By SSS-criteria]


V. RIGHT ANGLE-HYPOTENUSE-SIDE (RHS) CONGRUENCE CRITERION
Two right angles are congruent, if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other triangle.
In the given figure, ΔABC and ΔDEF are right-angled triangles in which Hyp. AC = Hyp. DF and BC = EF.

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

 

Ex.1 In ΔABC, AB = AC. If P is a point on AB and Q is a point on AC such that AP = AQ.
 Prove that
 (i) ΔAPC
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔAQB (ii) ΔBPC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔCQB.
 

Sol. Given : In ΔABC, AB = AC. P is a point on AB and Q is a point on AC such that AP = AQ.
To prove : (i) ΔAPC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ΔAQB

(ii) ΔBPC Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ΔCQB.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

 

Ex.2 In figure, diagonal AC of a quadrilateral ABCD bisects the angles A and C. Prove that AB = AD
 and CB = CD.

 

Sol. Given : In figure, diagonal AC of a quadrilateral ABCD bisects the angles A and C.

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

To prove : AB = AD and CB = CD

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

 

Ex. 3 AB is a line-segment. AX and BY are two equal line-segments drawn on opposite sides of line AB such that AX || BY. If AB and XY intersect each other at P.
 Prove that : 

(i) ΔAPX Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔBPY 

(ii) AB and XY bisect each other at P.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
 

Sol. Given : AB is a line-segment. AX and BY are two equal line-segments drawn on opposite
sides of line AB such that AX || BY.
To prove :

(i) ΔAPX Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔBPY

(ii) AB and XY bisect each other at P.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

 

Ex.4 In fig., AD is a median and BL, CM are perpendiculars drawn from B and C respectively on AD and AD produced. Prove that BL = CM.
 Sol. 
Given : In fig., AD is a median and BL, CM are perpendiculars drawn from B and C respectively on AD and AD produced. D

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9


Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

 

Ex.5 In fig. Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9BCD = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ADC and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ACB = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9BDA.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Prove that AD = BC and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B.
 Sol. 
Given : In fig. Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9BCD = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ADC and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ACB = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9BDA
To prove : (i) AD = BC (ii) Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Hence, proved.

 

Ex.6 In the fig. ABCD is a quadrilateral in which AB = AD and BC = DC. Prove that :
 (i) AC bisects each of the angles A and C.
 (ii) BE = ED.
 (iii) 
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ABC = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ADC. Can we say that AE = EC? 

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Sol. Given : In the fig. ABCD is a quadrilateral in which
AB = AD and BC = DC.
 

To prove : (i) AC bisects each of the angles A and C.
(ii) BE = ED.
(iii) Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ABC = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ADC. Can we say that AE = EC?

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9


No, AE Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 EC. For AE to be equal to EC, it is necessary that ΔAED Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔCED for which Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 92 must be equal to Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 94.
Hence, proved.

 

Ex. 7 ΔABC and ΔPBC are two isosceles triangles on the same base BC and vertices A and P are on the same side of BC.
 A and P are joined, show that :
 (i) ΔABP Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔACP and 

(ii) AP bisects Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A of ΔABC.
 

Sol. Given : ΔABC and ΔPBC are two isosceles triangles on the same base BC and vertices A and P are on
the same side of BC.
To prove : (i) ΔABP Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 ΔACP and (ii) AP bisects Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A of ΔABC.
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

 

Ex.8 AD, BE and CF, the altitudes of ΔABC are equal. Prove that ΔABC is an equilateral triangle.

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Sol. Given : AD, BE and CF are the altitudes of ABC and are equal. AP = BE = CF
To prove : ΔABC is an equilateral triangle

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

 

Ex. 9 In a ΔABC, the internal bisectors of Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9C meet at O. Prove that OA is the internal bisector of Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A.

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Sol. Given : In a ΔABC, the internal bisectors of Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B and Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9C meet at O.
To prove : OA bisects Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A.
Construction : Draw OD Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 BC, OE Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 CA and OFChapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9  AB.
Proof :

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Hence, proved.


PROPERTIES OF AN ISOSCELES TRIANGLE
In this section, we will learn some properties related to a triangle whose two sides are equal. We know that a triangle whose two sides are equal is called an isosceles triangle. Here, we will apply SAS congruence criteria and ASA (or AAS) congruence criteria to study some properties of an isosceles triangle.

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
THEOREM-3 : Angles opposite to equal sides of an isosceles triangle are equal.
Given : ΔABC is an isosceles triangle and AB = AC.
To prove :Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9C.
Construction : Draw AD the bisector of Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A. AD meets BC at D.
Proof :
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Hence, proved.

COROLLARY : Each angle of an equilateral triangle is of 60°.

Given : ΔABC is an equilateral triangle.
To prove : Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9A = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9C = 60°.
Proof :
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
 

THEOREM-4 : The sides opposite to equal angles of a triangle are equal.
 Given : 
ΔABC in which Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9B = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9C.

Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
To prove : AB = AC.
Construction : Draw AD Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 BC. AD meets BC in D.
Proof :
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Hence, proved.


Ex.10 In the adjoining fig, find the value of x. 
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
 

Sol. We have,
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9CAD + Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ADC + Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9DCA = 180° [Angle sum property]
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
But CD = CA ⇒ CAD = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ADC            [Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9s opposite to equal sides of a Δ are equal]
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9
Now, Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ADC = Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9ABD + Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9DAB [Ext. Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 of a Δ = sum of int. opp. Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9s]
Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9

The document Chapter 7 - Triangles, Solved Examples, Class 9, Maths | Extra Documents & Tests for Class 9 is a part of the Class 9 Course Extra Documents & Tests for Class 9.
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FAQs on Chapter 7 - Triangles, Solved Examples, Class 9, Maths - Extra Documents & Tests for Class 9

1. What are the different types of triangles?
Ans. There are three types of triangles based on their sides- Equilateral, Isosceles and Scalene. There are also three types of triangles based on their angles- Acute, Right-angled, and Obtuse.
2. How do we find the area of a triangle?
Ans. The area of a triangle is calculated by the formula, Area = 1/2 × base × height, where base is the length of the base of the triangle and height is the perpendicular distance from the base to the opposite vertex.
3. How do we find the perimeter of a triangle?
Ans. The perimeter of a triangle is calculated by adding the lengths of its three sides.
4. What is Pythagoras theorem?
Ans. Pythagoras theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
5. How do we determine if two triangles are congruent?
Ans. Two triangles are congruent if all three sides and all three angles of one triangle are equal to the corresponding sides and angles of the other triangle. We can use the congruency rules like SSS (side-side-side), SAS (side-angle-side), ASA (angle-side-angle), RHS (right angle-hypotenuse-side) to prove the congruency of two triangles.
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