Topics
Integers
 Concept for Natural Numbers
 Concept for Whole Numbers
 Concept of Negative Numbers
 Concept of Integers
 Representation of Integers on the Number Line
 Concept for Ordering of Integers
 Addition of Integers
 Addition of Integers on Number line
 Subtraction of Integers
 Properties of Addition and Subtraction of Integers
 Multiplication of a Positive and a Negative Integers
 Multiplication of Two Negative Integers
 Product of Three Or More Negative Integers
 Closure Property of Multiplication of Integers
 Commutative Property of Multiplication of Integers
 Associative Property of Multiplication of Integers
 Distributive Property of Multiplication of Integers
 Multiplication of Integers with Zero
 Multiplicative Identity of Integers
 Making Multiplication Easier of Integers
 Division of Integers
 Properties of Division of Integers
Fractions and Decimals
 Concept of Fractions
 Types of Fraction
 Concept of Proper Fractions.
 Improper Fraction and Mixed Fraction
 Concept for Equivalent Fractions
 Like and Unlike Fraction
 Comparing Fractions
 Addition of Fraction
 Subtraction of Fraction
 Multiplication of a Fraction by a Whole Number
 Fraction as an Operator 'Of'
 Multiplication of a Fraction by a Fraction
 Division of Fractions
 Concept for Reciprocal of a Fraction
 Concept of Decimal Numbers
 Multiplication of Decimal Numbers
 Multiplication of Decimal Numbers by 10, 100 and 1000
 Division of Decimal Numbers by 10, 100 and 1000
 Division of a Decimal Number by a Whole Number
 Division of a Decimal Number by Another Decimal Number
Data Handling
Simple Equations
Lines and Angles
 Concept of Points
 Concept of Line
 Concept of Line Segment
 Concept of Intersecting Lines
 Concept of Angle  Arms, Vertex, Interior and Exterior Region
 Complementary Angles
 Supplementary Angles
 Adjacent Angles
 Concept of Linear Pair
 Concept of Vertically Opposite Angles
 Concept of Intersecting Lines
 Concept of Parallel Lines
 Pairs of Lines  Transversal
 Pairs of Lines  Angles Made by a Transversal
 Pairs of Lines  Transversal of Parallel Lines
 Checking Parallel Lines
The Triangle and Its Properties
 Concept of Triangles  Sides, Angles, Vertices, Interior and Exterior of Triangle
 Classification of Triangles (On the Basis of Sides, and of Angles)
 Equilateral Triangle
 Isosceles Triangles
 Scalene Triangle
 Acute Angled Triangle
 Obtuse Angled Triangle
 Right Angled Triangle
 Median of a Triangle
 Altitudes of a Triangle
 Exterior Angle of a Triangle and Its Property
 Angle Sum Property of a Triangle
 Some Special Types of Triangles  Equilateral and Isosceles Triangles
 Sum of the Lengths of Two Sides of a Triangle
 Rightangled Triangles and Pythagoras Property
Congruence of Triangles
Comparing Quantities
 Concept of Ratio
 Concept of Equivalent Ratios
 Concept of Proportion
 Concept of Unitary Method
 Concept of Percent and Percentage
 Converting Fractional Numbers to Percentage
 Converting Decimals to Percentage
 Converting Percentages to Fractions
 Converting Percentages to Decimals
 Estimation in Percentages
 Interpreting Percentages
 Converting Percentages to “How Many”
 Ratios to Percents
 Increase Or Decrease as Percent
 Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
 Profit or Loss as a Percentage
 Concept of Principal, Interest, Amount, and Simple Interest
Rational Numbers
 Concept of Rational Numbers
 Equivalent Rational Number
 Positive and Negative Rational Numbers
 Rational Numbers on a Number Line
 Rational Numbers in Standard Form
 Comparison of Rational Numbers
 Rational Numbers Between Two Rational Numbers
 Addition of Rational Number
 Subtraction of Rational Number
 Multiplication of Rational Numbers
 Division of Rational Numbers
Practical Geometry
 Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
 Construction of Triangles
 Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
 Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
 Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
 Constructing a Rightangled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)
Perimeter and Area
 Mensuration
 Concept of Perimeter
 Perimeter of a Rectangle
 Perimeter of Squares
 Perimeter of Triangles
 Perimeter of Polygon
 Concept of Area
 Area of Square
 Area of Rectangle
 Triangles as Parts of Rectangles and Square
 Generalising for Other Congruent Parts of Rectangles
 Area of a Triangle
 Area of a Parallelogram
 Circumference of a Circle
 Area of Circle
 Conversion of Units
 Problems based on Perimeter and Area
 Problems based on Perimeter and Area
Algebraic Expressions
 Algebraic Expressions
 Terms, Factors and Coefficients of Expression
 Like and Unlike Terms
 Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
 Addition of Algebraic Expressions
 Subtraction of Algebraic Expressions
 Evaluation of Algebraic Expressions by Substituting a Value for the Variable.
 Use of Variables in Common Rules
Exponents and Powers
 Concept of Exponents
 Multiplying Powers with the Same Base
 Dividing Powers with the Same Base
 Taking Power of a Power
 Multiplying Powers with Different Base and Same Exponents
 Dividing Powers with Different Base and Same Exponents
 Numbers with Exponent Zero, One, Negative Exponents
 Miscellaneous Examples Using the Laws of Exponents
 Decimal Number System Using Exponents and Powers
 Expressing Large Numbers in the Standard Form
Symmetry
Visualizing Solid Shapes
 Plane Figures and Solid Shapes
 Faces, Edges and Vertices
 Nets for Building 3d Shapes  Cube, Cuboids, Cylinders, Cones, Pyramid, and Prism
 Drawing Solids on a Flat Surface  Oblique Sketches
 Drawing Solids on a Flat Surface  Isometric Sketches
 Visualising Solid Objects
 Viewing Different Sections of a Solid
definition
Vertically Opposite Angles: When two lines intersect we have two pairs of opposite angles. They are called vertically opposite angles.
Point of intersection: The meeting point of two lines is called the point of intersection.
notes
Vertically Opposite Angles:
 When two lines intersect we have two pairs of opposite angles. They are called vertically opposite angles.
 When two lines l and m meet, we say they intersect; the meeting point is called the point of intersection.

If two lines intersect each other, then the vertically opposite angles are equal.

If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are Obtuse angles.

Adjacent angles are linear and Supplementary angles i.e.,
∠AOC + ∠AOD = 180°.
∠AOD + ∠DOB = 180°.
∠DOB + ∠BOC = 180°.
∠BOC + ∠AOC = 180°.
theorem
When two lines intersect, the vertically opposite angles so formed are equal.
To Prove: ∠PQR and ∠SQT are congruent angles.
Proof:
As shown in the figure, m∠PQS = a, m∠SQT = b, m∠TQR = c, m∠PQR = d.
∠PQS and ∠SQT are the angles in a linear pair.
∴ a + b = 180°
Also, m∠SQT and m∠TQR are two angles in a linear pair.
∴ b + c = 180°
∴ a + b = b + c
∴ a = c.........(Subtracting b from both sides.)
∴ ∠PQS and ∠TQR are congruent angles.
Also, m∠PQR = m∠SQT
i.e. ∠PQR and ∠SQT are congruent angles.
Hence Proved.
Example
In the given figure, Identify two pairs of vertically opposite angles.
Vertically opposite angles are: (∠COB, ∠AOD), and (∠AOC, ∠BOD)