Topics
Number System
Rational Numbers
- Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
- Rational Numbers on a Number Line
- Inserting Rational Numbers Between Two Given Rational Numbers
- Method of Finding a Large Number of Rational Numbers Between Two Given Rational Numbers
Exponents
- Concept of Exponents
- Law of Exponents (For Integral Powers)
- Negative Integral Exponents
- More About Exponents
Squares and Square Root
- Concept of Square Roots
- Finding Square Root by Division Method
- Finding Square Root Through Prime Factorisation
- To Find the Square Root of a Number Which is Not a Perfect Square (Using Division Method)
- Properties of Square Numbers
Cubes and Cube Roots
Playing with Numbers
- Arranging the Objects in Rows and Columns
- Generalized Form of Numbers
- Some Interesting Properties
- Letters for Digits (Cryptarithms)
- Divisibility by 10
- Divisibility by 2
- Divisibility by 5
- Divisibility by 3
- Divisibility by 6
- Divisibility by 11
- Divisibility by 4
Sets
- Concept of Sets
- Representation of a Set
- Cardinality of a Set
- Types of Sets
- Subset
- Proper Subset
- Super Set
- Universal Set
- Complement of a Set
- Difference of Two Sets
- Distributive Laws
- Venn Diagrams
Ratio and Proportion
Percent and Percentage
Profit, Loss and Discount
- Overhead Expenses
- To Find S.P., When C.P. and Gain (Or Loss) Percent Are Given
- To Find C.P., When S.P. and Gain (Or Loss) Percent Are Given
- Concept of Discount
Interest
- Calculation of Interest
- Concept of Compound Interest
- To Find the Principle (P); the Rate Percent (R) and the Time
- Interest Compounded Half Yearly
- Applications of Compound Interest Formula
Direct and Inverse Variations
- Types of Variation
- Unitary Method
- Concept of Arrow Method
- Time and Work
Algebra
Algebraic Expressions
- Algebraic Expressions
- Degree of Polynomial
- Factors and Common Factors
- Classification of Terms in Algebra
- Combining like Terms
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Polynomial
- Multiplying a Polynomial by a Polynomial
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Simplification of Expressions
Identities
- Algebraic Identities
- Product of Sum and Difference of Two Terms
- Expansion Form of Numbers
- Important Formula of Expansion
- Cubes of Binomials
- Application of Formulae
Factorisation
- Factorisation by Taking Out Common Factors
- Factorisation by Taking Out Common Factors
- Factorisation by Grouping
- Factorisation by Difference of Two Squares
- Factorisation of Trinomials
- Factorising a Perfect Square Trinomial
- Factorising Completely
Linear Equations in One Variable
Linear Inequations
- Pair of Linear Equations in Two Variables
- Replacement Set and Solution Set
- Operation of Whole Numbers on Number Line
- Concept of Properties
Geometry
Understanding Shapes
Special Types of Quadrilaterals
Constructions
- Introduction of Constructions
- Construction of an Angle
- To Construct an Angle Equal to Given Angle
- To Draw the Bisector of a Given Angle
- Construction of an Angle Bisector: 30°, 45°, 60°, 90°
- Construction of Bisector of a Line
- The Perpendicular Bisector
- Construction of Parallel Lines
- Constructing a Quadrilateral
- Construction of Parallelograms
- Construction of a Rectangle When Its Length and Breadth Are Given.
- Construction of Rhombus
- Square: Properties and Construction
- Reflection Symmetry (Mirror Symmetry)
Representing 3-D in 2-D
- 2dimensional Perspective of 3dimensional Objects
- Concept of Polyhedron
- Faces, Edges and Vertices of Polyhedron
- Euler's Formula
- Concept of Polyhedron
- Nets of 3D Figures
Mensuration
Area of a Trapezium and a Polygon
Surface Area, Volume and Capacity
Data Handling (Statistics)
Data Handling
- Mathematical Data Collection and Organisation
- Frequency
- Raw Data, Arrayed Data and Frequency Distribution
- Cumulative Frequency and Cumulative Frequency Table
- Frequency Distribution Table
- Class Intervals and Class Limits
- Frequency Distribution and Its Applications
Probability
- Introduction
- Drawing an Angle Using a Protractor
- Construct an Angle Equal to a Given Angle
- Constructing Particular Angles (Bisecting an Angle)
- Key Points Summary
Introduction
In geometry, constructing angles accurately is an essential skill.
Whether using a protractor or compass and ruler, angle construction is used in practical fields like engineering, design, drafting, and architecture.
It allows us to draw angles of specific measures or copy and bisect them without needing advanced tools.
Drawing an Angle Using a Protractor
- Step 1: Draw a base line \[\overrightarrow{IN}\]
- Step 2: We will place the center point of the protractor on I and align IN to the 0 line.
- Step 3: Mark the required angle on the protractor scale. Mark point T at the label 30°.
- Step 4: Using a ruler, join the points I and T.
∠TIN = 30° is the required angle.
Construct an Angle Equal to a Given Angle
1. Draw ray QR.
2. Place the compass point at vertex B of ∠ABC and, taking a convenient distance, draw an arc to cut the rays BA and BC at points D and E, respectively.
3. Using the same distance again, place the compass point at point Q of ray QR and draw an arc. Let this arc cut the ray QR at T.
4. Now place the point of the compass at point E and open the compass to a distance equal to DE.
5. Now, align the compass point with point T.
6. Using the distance equal to DE, draw an arc to cut the previous arc at S.
7. Draw the ray QS. Take any point P on ray QS.
8. Using a protractor, verify that the ∠PQR thus formed is of the same measure as ∠ABC.
Constructing Particular Angles (Bisecting an Angle)
To construct an angle of 60°.
Steps:
-
Draw Base Ray: Draw a ray OA.
- First Arc: With O as the center, draw an arc of any suitable radius that cuts OA at point B.
-
Second Arc: Without changing the radius, set the compass at B and draw an arc that cuts the first arc at point C.
- Join OC and extend to a suitable point D.
Then, ∠DOA = 60°.
To construct an angle of 30°.
Steps:
- Draw an angle AOB of 60°, as explained above.
- Bisect this angle to get two angles of 30° each.
Thus, ∠EOB = 30°.
Key Points Summary
-
Use a protractor to draw angles directly.
-
Use a compass and ruler to copy or construct angles without measuring.
-
You can bisect angles to get smaller standard angles.
-
Many angles are built using combinations of 30°, 45°, 60°, and 90°.
