Syllabus: Mathematics for Class 9 | Mathematics (Maths) Class 9 PDF Download

 Page 1


 
COURSE STRUCTURE CLASS – IX 
 
Units Unit Name Marks 
I NUMBER SYSTEMS 10 
II ALGEBRA 20 
III COORDINATE GEOMETRY 04 
IV GEOMETRY 27 
V MENSURATION 13 
VI STATISTICS 06 
   Total 80 
 
S. 
No. 
Content Competencies Explanation 
 
Unit 1: Number Systems 
 
1. REAL NUMBERS 
 
1. Review of representation of natural 
numbers, integers, rational numbers 
on the number line. Representation of 
terminating/non-terminating recurring 
decimals on the number line through 
successive magnification, Rational 
numbers as recurring/ terminating 
decimals. Operations on real 
numbers. 
2. Examples of non-recurring/non-
terminating decimals. Existence of 
non-rational numbers (irrational 
numbers) such as 
v
2, v3  and their 
representation on the number line. 
Explaining that every real number is 
represented by a unique point on the 
number line and conversely, viz. every 
point on the number line represents a 
unique real number.  
3. Definition of nth root of a real number.  
4. Rationalization (with precise 
meaning) of real numbers of the type  
1
?? +?? v??  and 
1
v?? +
v
??  (and their 
combinations), where ??  and ??  are 
natural numbers and ??  and ??  are 
integers.  
? Develops a deeper 
understanding of 
numbers, including 
the set of real 
numbers and its 
properties. 
? Recognizes and 
appropriately uses 
powers and 
exponents. 
? Computes powers 
and roots and 
applies them to 
solve problems. 
 
? Differentiates 
rational and 
irrational numbers 
based on decimal 
representation. 
? Represents 
rational and 
irrational numbers 
on the number line. 
? Rationalizes real 
number 
expressions such 
as 
1
?? +?? v??  and 
1
v?? +
v
?? , where x, y 
are natural 
numbers and a, b 
are integers. 
? Applies laws of 
exponents 
Page 2


 
COURSE STRUCTURE CLASS – IX 
 
Units Unit Name Marks 
I NUMBER SYSTEMS 10 
II ALGEBRA 20 
III COORDINATE GEOMETRY 04 
IV GEOMETRY 27 
V MENSURATION 13 
VI STATISTICS 06 
   Total 80 
 
S. 
No. 
Content Competencies Explanation 
 
Unit 1: Number Systems 
 
1. REAL NUMBERS 
 
1. Review of representation of natural 
numbers, integers, rational numbers 
on the number line. Representation of 
terminating/non-terminating recurring 
decimals on the number line through 
successive magnification, Rational 
numbers as recurring/ terminating 
decimals. Operations on real 
numbers. 
2. Examples of non-recurring/non-
terminating decimals. Existence of 
non-rational numbers (irrational 
numbers) such as 
v
2, v3  and their 
representation on the number line. 
Explaining that every real number is 
represented by a unique point on the 
number line and conversely, viz. every 
point on the number line represents a 
unique real number.  
3. Definition of nth root of a real number.  
4. Rationalization (with precise 
meaning) of real numbers of the type  
1
?? +?? v??  and 
1
v?? +
v
??  (and their 
combinations), where ??  and ??  are 
natural numbers and ??  and ??  are 
integers.  
? Develops a deeper 
understanding of 
numbers, including 
the set of real 
numbers and its 
properties. 
? Recognizes and 
appropriately uses 
powers and 
exponents. 
? Computes powers 
and roots and 
applies them to 
solve problems. 
 
? Differentiates 
rational and 
irrational numbers 
based on decimal 
representation. 
? Represents 
rational and 
irrational numbers 
on the number line. 
? Rationalizes real 
number 
expressions such 
as 
1
?? +?? v??  and 
1
v?? +
v
?? , where x, y 
are natural 
numbers and a, b 
are integers. 
? Applies laws of 
exponents 
5. Recall of laws of exponents with 
integral powers. Rational exponents 
with positive real bases (to be done by 
particular cases, allowing learner to 
arrive at the general laws.) 
 
UNIT II: ALGEBRA 
 
1. POLYNOMIALS  
 
1. Definition of a polynomial in one 
variable, with examples and counter 
examples. Coefficients of a 
polynomial, terms of a polynomial 
and zero polynomial.  
2. Degree of a polynomial. 
3. Constant, linear, quadratic and cubic 
polynomials. Monomials, binomials, 
trinomials. Factors and multiples. 
4. Zeroes of a polynomial. 
5. Motivate and State the Remainder 
Theorem with examples.  
6. Statement and proof of the Factor 
Theorem. Factorization of ax
2
 + bx + 
c, a ? 0 where a, b and c are real 
numbers, and of cubic polynomials 
using the Factor theorem.  
7. Recall of algebraic expressions and 
identities. Verification of identities:  
(?? + ?? + ?? )
2
= ?? 2
+ ?? 2
+ ?? 2
+ 2????
+ 2???? + 2???? 
(?? ± ?? )
3
= ?? 3
± ?? 3
± 3???? (?? ± ?? ) 
?? 3
+ ?? 3
= (?? + ?? )(?? 2
- ???? + ?? 2
) 
?? 3
- ?? 3
= (?? - ?? )(?? 2
+ ???? + ?? 2
 
?? 3
+ ?? 3
+ ?? 3
- 3?????? = (?? + ?? + ?? )(?? 2
+ ?? 2
+ ?? 2
- ???? - ???? - ???? ) 
  and their use in factorization 
 of polynomials. 
 
? Learns the art of 
factoring 
polynomials. 
 
? Defines 
polynomials in 
one variable. 
? Identifies different 
terms and 
different types of 
polynomials. 
? Finds zeros of a 
polynomial 
? Proves factor 
theorem and 
applies the 
theorem to 
factorize 
polynomials. 
? Proves and 
applies algebraic 
identities up to 
degree three. 
 
 
2. LINEAR EQUATIONS IN TWO 
VARIABLES 
 
1. Recall of linear equations in one 
variable.  
2. Introduction to the equation in two 
variables. Focus on linear equations 
of the type ax + by + c = 0.             
? Visualizes solutions 
of a linear equation 
in two variables as 
ordered pair of real 
numbers on its 
graph 
? Describes and plot 
a linear equation in 
two variables. 
 
 
Page 3


 
COURSE STRUCTURE CLASS – IX 
 
Units Unit Name Marks 
I NUMBER SYSTEMS 10 
II ALGEBRA 20 
III COORDINATE GEOMETRY 04 
IV GEOMETRY 27 
V MENSURATION 13 
VI STATISTICS 06 
   Total 80 
 
S. 
No. 
Content Competencies Explanation 
 
Unit 1: Number Systems 
 
1. REAL NUMBERS 
 
1. Review of representation of natural 
numbers, integers, rational numbers 
on the number line. Representation of 
terminating/non-terminating recurring 
decimals on the number line through 
successive magnification, Rational 
numbers as recurring/ terminating 
decimals. Operations on real 
numbers. 
2. Examples of non-recurring/non-
terminating decimals. Existence of 
non-rational numbers (irrational 
numbers) such as 
v
2, v3  and their 
representation on the number line. 
Explaining that every real number is 
represented by a unique point on the 
number line and conversely, viz. every 
point on the number line represents a 
unique real number.  
3. Definition of nth root of a real number.  
4. Rationalization (with precise 
meaning) of real numbers of the type  
1
?? +?? v??  and 
1
v?? +
v
??  (and their 
combinations), where ??  and ??  are 
natural numbers and ??  and ??  are 
integers.  
? Develops a deeper 
understanding of 
numbers, including 
the set of real 
numbers and its 
properties. 
? Recognizes and 
appropriately uses 
powers and 
exponents. 
? Computes powers 
and roots and 
applies them to 
solve problems. 
 
? Differentiates 
rational and 
irrational numbers 
based on decimal 
representation. 
? Represents 
rational and 
irrational numbers 
on the number line. 
? Rationalizes real 
number 
expressions such 
as 
1
?? +?? v??  and 
1
v?? +
v
?? , where x, y 
are natural 
numbers and a, b 
are integers. 
? Applies laws of 
exponents 
5. Recall of laws of exponents with 
integral powers. Rational exponents 
with positive real bases (to be done by 
particular cases, allowing learner to 
arrive at the general laws.) 
 
UNIT II: ALGEBRA 
 
1. POLYNOMIALS  
 
1. Definition of a polynomial in one 
variable, with examples and counter 
examples. Coefficients of a 
polynomial, terms of a polynomial 
and zero polynomial.  
2. Degree of a polynomial. 
3. Constant, linear, quadratic and cubic 
polynomials. Monomials, binomials, 
trinomials. Factors and multiples. 
4. Zeroes of a polynomial. 
5. Motivate and State the Remainder 
Theorem with examples.  
6. Statement and proof of the Factor 
Theorem. Factorization of ax
2
 + bx + 
c, a ? 0 where a, b and c are real 
numbers, and of cubic polynomials 
using the Factor theorem.  
7. Recall of algebraic expressions and 
identities. Verification of identities:  
(?? + ?? + ?? )
2
= ?? 2
+ ?? 2
+ ?? 2
+ 2????
+ 2???? + 2???? 
(?? ± ?? )
3
= ?? 3
± ?? 3
± 3???? (?? ± ?? ) 
?? 3
+ ?? 3
= (?? + ?? )(?? 2
- ???? + ?? 2
) 
?? 3
- ?? 3
= (?? - ?? )(?? 2
+ ???? + ?? 2
 
?? 3
+ ?? 3
+ ?? 3
- 3?????? = (?? + ?? + ?? )(?? 2
+ ?? 2
+ ?? 2
- ???? - ???? - ???? ) 
  and their use in factorization 
 of polynomials. 
 
? Learns the art of 
factoring 
polynomials. 
 
? Defines 
polynomials in 
one variable. 
? Identifies different 
terms and 
different types of 
polynomials. 
? Finds zeros of a 
polynomial 
? Proves factor 
theorem and 
applies the 
theorem to 
factorize 
polynomials. 
? Proves and 
applies algebraic 
identities up to 
degree three. 
 
 
2. LINEAR EQUATIONS IN TWO 
VARIABLES 
 
1. Recall of linear equations in one 
variable.  
2. Introduction to the equation in two 
variables. Focus on linear equations 
of the type ax + by + c = 0.             
? Visualizes solutions 
of a linear equation 
in two variables as 
ordered pair of real 
numbers on its 
graph 
? Describes and plot 
a linear equation in 
two variables. 
 
 
Explain that a linear equation in two 
variables has infinitely many 
solutions and justify their being 
written as ordered pairs of real 
numbers, plotting them and showing 
that they lie on a line.  
 
UNIT III: COORDINATE GEOMETRY 
 
1. Coordinate Geometry: 
 
1. The Cartesian plane, coordinates of 
a point 
2. Names and terms associated with the 
coordinate plane, notations. 
? Specifies locations 
and describes 
spatial relationships 
using coordinate 
geometry. 
? Describes 
cartesian plane 
and its 
associated terms 
and notations 
 
 
UNIT IV: GEOMETRY 
 
1. INTRODUCTION TO EUCLID’S 
GEOMETRY 
 
1. History - Geometry in India and 
Euclid's geometry. Euclid's method of 
formalizing observed phenomenon 
into rigorous Mathematics with 
definitions, common/obvious notions, 
axioms/postulates and theorems.  
2. The five postulates of Euclid. 
Equivalent versions of the fifth 
postulate. Showing the relationship 
between axiom and theorem, for 
example:  
(a) Given two distinct points, there exists 
one and only one line through them. 
(Axiom)  
(b) (Prove) Two distinct lines cannot 
have more than one point in 
common. (Theorem) 
  
? Proves theorems 
using Euclid’s 
axioms and 
postulates– for 
triangles, 
quadrilaterals, and 
circles and applies 
them to solve 
geometric 
problems. 
 
 
? Understands 
historical 
relevance of Indian 
and Euclidean 
Geometry. 
? Defines axioms, 
postulates, 
theorems with 
reference to 
Euclidean 
Geometry. 
 
 
 
 
 
 
 
2. LINES AND ANGLES 
 
1. (State without proof) If a ray stands on 
a line, then the sum of the two 
adjacent angles so formed is 180° 
and the converse. 
2. (Prove) If two lines intersect, vertically 
opposite angles are equal.  
3. (State without proof) Lines which are 
parallel to a given line are parallel.  
 
? derives proofs of 
mathematical 
statements 
particularly related to 
geometrical concepts, 
like parallel lines by 
applying axiomatic 
approach and solves 
problems using them. 
? Visualizes, 
explains and 
applies relations 
between different 
pairs of angles on 
a set of parallel 
lines and 
intersecting 
transversal. 
Page 4


 
COURSE STRUCTURE CLASS – IX 
 
Units Unit Name Marks 
I NUMBER SYSTEMS 10 
II ALGEBRA 20 
III COORDINATE GEOMETRY 04 
IV GEOMETRY 27 
V MENSURATION 13 
VI STATISTICS 06 
   Total 80 
 
S. 
No. 
Content Competencies Explanation 
 
Unit 1: Number Systems 
 
1. REAL NUMBERS 
 
1. Review of representation of natural 
numbers, integers, rational numbers 
on the number line. Representation of 
terminating/non-terminating recurring 
decimals on the number line through 
successive magnification, Rational 
numbers as recurring/ terminating 
decimals. Operations on real 
numbers. 
2. Examples of non-recurring/non-
terminating decimals. Existence of 
non-rational numbers (irrational 
numbers) such as 
v
2, v3  and their 
representation on the number line. 
Explaining that every real number is 
represented by a unique point on the 
number line and conversely, viz. every 
point on the number line represents a 
unique real number.  
3. Definition of nth root of a real number.  
4. Rationalization (with precise 
meaning) of real numbers of the type  
1
?? +?? v??  and 
1
v?? +
v
??  (and their 
combinations), where ??  and ??  are 
natural numbers and ??  and ??  are 
integers.  
? Develops a deeper 
understanding of 
numbers, including 
the set of real 
numbers and its 
properties. 
? Recognizes and 
appropriately uses 
powers and 
exponents. 
? Computes powers 
and roots and 
applies them to 
solve problems. 
 
? Differentiates 
rational and 
irrational numbers 
based on decimal 
representation. 
? Represents 
rational and 
irrational numbers 
on the number line. 
? Rationalizes real 
number 
expressions such 
as 
1
?? +?? v??  and 
1
v?? +
v
?? , where x, y 
are natural 
numbers and a, b 
are integers. 
? Applies laws of 
exponents 
5. Recall of laws of exponents with 
integral powers. Rational exponents 
with positive real bases (to be done by 
particular cases, allowing learner to 
arrive at the general laws.) 
 
UNIT II: ALGEBRA 
 
1. POLYNOMIALS  
 
1. Definition of a polynomial in one 
variable, with examples and counter 
examples. Coefficients of a 
polynomial, terms of a polynomial 
and zero polynomial.  
2. Degree of a polynomial. 
3. Constant, linear, quadratic and cubic 
polynomials. Monomials, binomials, 
trinomials. Factors and multiples. 
4. Zeroes of a polynomial. 
5. Motivate and State the Remainder 
Theorem with examples.  
6. Statement and proof of the Factor 
Theorem. Factorization of ax
2
 + bx + 
c, a ? 0 where a, b and c are real 
numbers, and of cubic polynomials 
using the Factor theorem.  
7. Recall of algebraic expressions and 
identities. Verification of identities:  
(?? + ?? + ?? )
2
= ?? 2
+ ?? 2
+ ?? 2
+ 2????
+ 2???? + 2???? 
(?? ± ?? )
3
= ?? 3
± ?? 3
± 3???? (?? ± ?? ) 
?? 3
+ ?? 3
= (?? + ?? )(?? 2
- ???? + ?? 2
) 
?? 3
- ?? 3
= (?? - ?? )(?? 2
+ ???? + ?? 2
 
?? 3
+ ?? 3
+ ?? 3
- 3?????? = (?? + ?? + ?? )(?? 2
+ ?? 2
+ ?? 2
- ???? - ???? - ???? ) 
  and their use in factorization 
 of polynomials. 
 
? Learns the art of 
factoring 
polynomials. 
 
? Defines 
polynomials in 
one variable. 
? Identifies different 
terms and 
different types of 
polynomials. 
? Finds zeros of a 
polynomial 
? Proves factor 
theorem and 
applies the 
theorem to 
factorize 
polynomials. 
? Proves and 
applies algebraic 
identities up to 
degree three. 
 
 
2. LINEAR EQUATIONS IN TWO 
VARIABLES 
 
1. Recall of linear equations in one 
variable.  
2. Introduction to the equation in two 
variables. Focus on linear equations 
of the type ax + by + c = 0.             
? Visualizes solutions 
of a linear equation 
in two variables as 
ordered pair of real 
numbers on its 
graph 
? Describes and plot 
a linear equation in 
two variables. 
 
 
Explain that a linear equation in two 
variables has infinitely many 
solutions and justify their being 
written as ordered pairs of real 
numbers, plotting them and showing 
that they lie on a line.  
 
UNIT III: COORDINATE GEOMETRY 
 
1. Coordinate Geometry: 
 
1. The Cartesian plane, coordinates of 
a point 
2. Names and terms associated with the 
coordinate plane, notations. 
? Specifies locations 
and describes 
spatial relationships 
using coordinate 
geometry. 
? Describes 
cartesian plane 
and its 
associated terms 
and notations 
 
 
UNIT IV: GEOMETRY 
 
1. INTRODUCTION TO EUCLID’S 
GEOMETRY 
 
1. History - Geometry in India and 
Euclid's geometry. Euclid's method of 
formalizing observed phenomenon 
into rigorous Mathematics with 
definitions, common/obvious notions, 
axioms/postulates and theorems.  
2. The five postulates of Euclid. 
Equivalent versions of the fifth 
postulate. Showing the relationship 
between axiom and theorem, for 
example:  
(a) Given two distinct points, there exists 
one and only one line through them. 
(Axiom)  
(b) (Prove) Two distinct lines cannot 
have more than one point in 
common. (Theorem) 
  
? Proves theorems 
using Euclid’s 
axioms and 
postulates– for 
triangles, 
quadrilaterals, and 
circles and applies 
them to solve 
geometric 
problems. 
 
 
? Understands 
historical 
relevance of Indian 
and Euclidean 
Geometry. 
? Defines axioms, 
postulates, 
theorems with 
reference to 
Euclidean 
Geometry. 
 
 
 
 
 
 
 
2. LINES AND ANGLES 
 
1. (State without proof) If a ray stands on 
a line, then the sum of the two 
adjacent angles so formed is 180° 
and the converse. 
2. (Prove) If two lines intersect, vertically 
opposite angles are equal.  
3. (State without proof) Lines which are 
parallel to a given line are parallel.  
 
? derives proofs of 
mathematical 
statements 
particularly related to 
geometrical concepts, 
like parallel lines by 
applying axiomatic 
approach and solves 
problems using them. 
? Visualizes, 
explains and 
applies relations 
between different 
pairs of angles on 
a set of parallel 
lines and 
intersecting 
transversal. 
? Solves problems 
based on parallel 
lines and 
intersecting 
transversal. 
 
3. TRIANGLES 
 
1. (State without proof) Two triangles are 
congruent if any two sides and the 
included angle of one triangle is equal 
(respectively) to any two sides and the 
included angle of the other triangle 
(SAS Congruence).  
2. (Prove) Two triangles are congruent if 
any two angles and the included side 
of one triangle is equal (respectively) 
to any two angles and the included 
side of the other triangle (ASA 
Congruence). 
3. (State without proof) Two triangles are 
congruent if the three sides of one 
triangle are equal (respectively) to 
three sides of the other triangle (SSS 
Congruence).  
4. (State without proof) Two right 
triangles are congruent if the 
hypotenuse and a side of one triangle 
are equal (respectively) to the 
hypotenuse and a side of the other 
triangle. (RHS Congruence). 
5. (Prove) The angles opposite to equal 
sides of a triangle are equal.  
6. (State without proof) The sides 
opposite to equal angles of a triangle 
are equal.  
 
? Describe 
relationships 
including 
congruency of two-
dimensional 
geometrical shapes 
(lines, angle, 
triangles) to make 
and test 
conjectures and 
solve problems. 
 
? derives proofs of 
mathematical 
statements 
particularly related 
to geometrical 
concepts triangles 
by applying 
axiomatic approach 
and solves 
problems using 
them. 
? Visualizes and 
explains 
congruence 
properties of two 
triangles. 
? Applies 
congruency criteria 
to solve problems 
 
4. QUADRILATERALS 
 
1. (Prove) The diagonal divides a 
parallelogram into two congruent 
triangles. 
2. (State without proof) In a 
parallelogram opposite sides are 
equal, and conversely.  
3. (State without proof) In a 
parallelogram opposite angles are 
equal, and conversely.  
? derives proofs of 
mathematical 
statements 
particularly related 
to geometrical 
concepts of 
quadrilaterals by 
applying axiomatic 
approach and 
solves problems 
using them. 
? Visualizes and 
explains 
properties of 
quadrilaterals 
? Solves problems 
based on 
properties of 
quadrilaterals. 
 
Page 5


 
COURSE STRUCTURE CLASS – IX 
 
Units Unit Name Marks 
I NUMBER SYSTEMS 10 
II ALGEBRA 20 
III COORDINATE GEOMETRY 04 
IV GEOMETRY 27 
V MENSURATION 13 
VI STATISTICS 06 
   Total 80 
 
S. 
No. 
Content Competencies Explanation 
 
Unit 1: Number Systems 
 
1. REAL NUMBERS 
 
1. Review of representation of natural 
numbers, integers, rational numbers 
on the number line. Representation of 
terminating/non-terminating recurring 
decimals on the number line through 
successive magnification, Rational 
numbers as recurring/ terminating 
decimals. Operations on real 
numbers. 
2. Examples of non-recurring/non-
terminating decimals. Existence of 
non-rational numbers (irrational 
numbers) such as 
v
2, v3  and their 
representation on the number line. 
Explaining that every real number is 
represented by a unique point on the 
number line and conversely, viz. every 
point on the number line represents a 
unique real number.  
3. Definition of nth root of a real number.  
4. Rationalization (with precise 
meaning) of real numbers of the type  
1
?? +?? v??  and 
1
v?? +
v
??  (and their 
combinations), where ??  and ??  are 
natural numbers and ??  and ??  are 
integers.  
? Develops a deeper 
understanding of 
numbers, including 
the set of real 
numbers and its 
properties. 
? Recognizes and 
appropriately uses 
powers and 
exponents. 
? Computes powers 
and roots and 
applies them to 
solve problems. 
 
? Differentiates 
rational and 
irrational numbers 
based on decimal 
representation. 
? Represents 
rational and 
irrational numbers 
on the number line. 
? Rationalizes real 
number 
expressions such 
as 
1
?? +?? v??  and 
1
v?? +
v
?? , where x, y 
are natural 
numbers and a, b 
are integers. 
? Applies laws of 
exponents 
5. Recall of laws of exponents with 
integral powers. Rational exponents 
with positive real bases (to be done by 
particular cases, allowing learner to 
arrive at the general laws.) 
 
UNIT II: ALGEBRA 
 
1. POLYNOMIALS  
 
1. Definition of a polynomial in one 
variable, with examples and counter 
examples. Coefficients of a 
polynomial, terms of a polynomial 
and zero polynomial.  
2. Degree of a polynomial. 
3. Constant, linear, quadratic and cubic 
polynomials. Monomials, binomials, 
trinomials. Factors and multiples. 
4. Zeroes of a polynomial. 
5. Motivate and State the Remainder 
Theorem with examples.  
6. Statement and proof of the Factor 
Theorem. Factorization of ax
2
 + bx + 
c, a ? 0 where a, b and c are real 
numbers, and of cubic polynomials 
using the Factor theorem.  
7. Recall of algebraic expressions and 
identities. Verification of identities:  
(?? + ?? + ?? )
2
= ?? 2
+ ?? 2
+ ?? 2
+ 2????
+ 2???? + 2???? 
(?? ± ?? )
3
= ?? 3
± ?? 3
± 3???? (?? ± ?? ) 
?? 3
+ ?? 3
= (?? + ?? )(?? 2
- ???? + ?? 2
) 
?? 3
- ?? 3
= (?? - ?? )(?? 2
+ ???? + ?? 2
 
?? 3
+ ?? 3
+ ?? 3
- 3?????? = (?? + ?? + ?? )(?? 2
+ ?? 2
+ ?? 2
- ???? - ???? - ???? ) 
  and their use in factorization 
 of polynomials. 
 
? Learns the art of 
factoring 
polynomials. 
 
? Defines 
polynomials in 
one variable. 
? Identifies different 
terms and 
different types of 
polynomials. 
? Finds zeros of a 
polynomial 
? Proves factor 
theorem and 
applies the 
theorem to 
factorize 
polynomials. 
? Proves and 
applies algebraic 
identities up to 
degree three. 
 
 
2. LINEAR EQUATIONS IN TWO 
VARIABLES 
 
1. Recall of linear equations in one 
variable.  
2. Introduction to the equation in two 
variables. Focus on linear equations 
of the type ax + by + c = 0.             
? Visualizes solutions 
of a linear equation 
in two variables as 
ordered pair of real 
numbers on its 
graph 
? Describes and plot 
a linear equation in 
two variables. 
 
 
Explain that a linear equation in two 
variables has infinitely many 
solutions and justify their being 
written as ordered pairs of real 
numbers, plotting them and showing 
that they lie on a line.  
 
UNIT III: COORDINATE GEOMETRY 
 
1. Coordinate Geometry: 
 
1. The Cartesian plane, coordinates of 
a point 
2. Names and terms associated with the 
coordinate plane, notations. 
? Specifies locations 
and describes 
spatial relationships 
using coordinate 
geometry. 
? Describes 
cartesian plane 
and its 
associated terms 
and notations 
 
 
UNIT IV: GEOMETRY 
 
1. INTRODUCTION TO EUCLID’S 
GEOMETRY 
 
1. History - Geometry in India and 
Euclid's geometry. Euclid's method of 
formalizing observed phenomenon 
into rigorous Mathematics with 
definitions, common/obvious notions, 
axioms/postulates and theorems.  
2. The five postulates of Euclid. 
Equivalent versions of the fifth 
postulate. Showing the relationship 
between axiom and theorem, for 
example:  
(a) Given two distinct points, there exists 
one and only one line through them. 
(Axiom)  
(b) (Prove) Two distinct lines cannot 
have more than one point in 
common. (Theorem) 
  
? Proves theorems 
using Euclid’s 
axioms and 
postulates– for 
triangles, 
quadrilaterals, and 
circles and applies 
them to solve 
geometric 
problems. 
 
 
? Understands 
historical 
relevance of Indian 
and Euclidean 
Geometry. 
? Defines axioms, 
postulates, 
theorems with 
reference to 
Euclidean 
Geometry. 
 
 
 
 
 
 
 
2. LINES AND ANGLES 
 
1. (State without proof) If a ray stands on 
a line, then the sum of the two 
adjacent angles so formed is 180° 
and the converse. 
2. (Prove) If two lines intersect, vertically 
opposite angles are equal.  
3. (State without proof) Lines which are 
parallel to a given line are parallel.  
 
? derives proofs of 
mathematical 
statements 
particularly related to 
geometrical concepts, 
like parallel lines by 
applying axiomatic 
approach and solves 
problems using them. 
? Visualizes, 
explains and 
applies relations 
between different 
pairs of angles on 
a set of parallel 
lines and 
intersecting 
transversal. 
? Solves problems 
based on parallel 
lines and 
intersecting 
transversal. 
 
3. TRIANGLES 
 
1. (State without proof) Two triangles are 
congruent if any two sides and the 
included angle of one triangle is equal 
(respectively) to any two sides and the 
included angle of the other triangle 
(SAS Congruence).  
2. (Prove) Two triangles are congruent if 
any two angles and the included side 
of one triangle is equal (respectively) 
to any two angles and the included 
side of the other triangle (ASA 
Congruence). 
3. (State without proof) Two triangles are 
congruent if the three sides of one 
triangle are equal (respectively) to 
three sides of the other triangle (SSS 
Congruence).  
4. (State without proof) Two right 
triangles are congruent if the 
hypotenuse and a side of one triangle 
are equal (respectively) to the 
hypotenuse and a side of the other 
triangle. (RHS Congruence). 
5. (Prove) The angles opposite to equal 
sides of a triangle are equal.  
6. (State without proof) The sides 
opposite to equal angles of a triangle 
are equal.  
 
? Describe 
relationships 
including 
congruency of two-
dimensional 
geometrical shapes 
(lines, angle, 
triangles) to make 
and test 
conjectures and 
solve problems. 
 
? derives proofs of 
mathematical 
statements 
particularly related 
to geometrical 
concepts triangles 
by applying 
axiomatic approach 
and solves 
problems using 
them. 
? Visualizes and 
explains 
congruence 
properties of two 
triangles. 
? Applies 
congruency criteria 
to solve problems 
 
4. QUADRILATERALS 
 
1. (Prove) The diagonal divides a 
parallelogram into two congruent 
triangles. 
2. (State without proof) In a 
parallelogram opposite sides are 
equal, and conversely.  
3. (State without proof) In a 
parallelogram opposite angles are 
equal, and conversely.  
? derives proofs of 
mathematical 
statements 
particularly related 
to geometrical 
concepts of 
quadrilaterals by 
applying axiomatic 
approach and 
solves problems 
using them. 
? Visualizes and 
explains 
properties of 
quadrilaterals 
? Solves problems 
based on 
properties of 
quadrilaterals. 
 
4. (State without proof) A quadrilateral is 
a parallelogram if a pair of its opposite 
sides is parallel and equal.  
5. (State without proof) In a 
parallelogram, the diagonals bisect 
each other and conversely.  
6. (State without proof) In a triangle, the 
line segment joining the mid points of 
any two sides is parallel to the third 
side and is half of it and (State without 
proof) its converse. 
 
 
5. CIRCLES 
 
1. (Prove) Equal chords of a circle 
subtend equal angles at the center 
and (State without proof) its converse.   
2. (State without proof) The 
perpendicular from the center of a 
circle to a chord bisects the chord and 
conversely, the line drawn through the 
center of a circle to bisect a chord is 
perpendicular to the chord.  
3. (State without proof) Equal chords of 
a circle (or of congruent circles) are 
equidistant from the center (or their 
respective centers) and conversely.  
4. (Prove) The angle subtended by an 
arc at the center is double the angle 
subtended by it at any point on the 
remaining part of the circle.  
5. (State without proof) Angles in the 
same segment of a circle are equal.  
6. (State without proof) If a line segment 
joining two points subtends equal 
angle at two other points lying on the 
same side of the line containing the 
segment, the four points lie on a 
circle.  
7. (State without proof) The sum of 
either of the pair of the opposite 
angles of a cyclic quadrilateral is 180° 
and its converse. 
 
 
 
? Proves 
theorems about 
the geometry of 
a circle, 
including its 
chords and 
subtended 
angles 
? Visualizes and 
explains properties 
of circles. 
? Solves problems 
based on 
properties of circle.