Topics
Number System(Consolidating the Sense of Numberness)
Number System
Estimation
Ratio and Proportion
Algebra
Numbers in India and International System (With Comparison)
Geometry
Place Value
Mensuration
Natural Numbers and Whole Numbers (Including Patterns)
Data Handling
Negative Numbers and Integers
Number Line
HCF and LCM
Playing with Numbers
- Simplification of Brackets
- Finding Factors Using Rectangular Arrangements and Division
- Factors and Common Factors
- Multiples and Common Multiples
- Concept of Even and Odd Number
- Tests for Divisibility of Numbers
- Divisibility by 2
- Divisibility by 4
- Divisibility by 8
- Divisibility by 3
- Divisibility by 6
- Divisibility by 9
- Divisibility by 5
- Divisibility by 11
Sets
Ratio
Proportion (Including Word Problems)
Unitary Method
Fractions
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Like and Unlike Fraction
- Concept of Equivalent Fractions
- Conversion between Improper and Mixed fraction
- Conversion between Unlike and Like Fractions
- Simplest Form of a Fractions
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of Fraction
- Division of Fractions
- Using Operator 'Of' with Multiplication and Division
- BODMAS Rule
- Problems Based on Fraction
Decimal Fractions
Percent (Percentage)
Idea of Speed, Distance and Time
Fundamental Concepts
Fundamental Operations (Related to Algebraic Expressions)
Substitution (Including Use of Brackets as Grouping Symbols)
Framing Algebraic Expressions (Including Evaluation)
Simple (Linear) Equations (Including Word Problems)
Fundamental Concepts
Angles (With Their Types)
Properties of Angles and Lines (Including Parallel Lines)
Triangles (Including Types, Properties and Constructions)
Quadrilateral
Polygons
The Circle
Symmetry (Including Constructions on Symmetry)
Recognition of Solids
Perimeter and Area of Plane Figures
Data Handling (Including Pictograph and Bar Graph)
Mean and Median
- Introduction
- Adjacent Angles
- Vertically Opposite Angles
- Straight Line Property
- Example 1
- Example 2
- Key Points Summary
Introduction
When two straight lines intersect, they form multiple angles. These angles follow certain geometric properties, which help us solve problems related to angle measurement and alignment. Understanding these properties is fundamental in geometry, construction, and design.
Adjacent Angles
Property: The sum of a pair of adjacent angles formed by two intersecting lines is always 180°.
Example:
The sum of adjacent angles = 180°.
i.e., ∠AOD + ∠DOB = 180°,
∠BOD + ∠BOC = 180°,
∠BOC + ∠COA = 180°,
and ∠COA + ∠AOD = 180°.
Vertically Opposite Angles
Property: Vertically opposite angles are always equal.
Example:
∠AOC = ∠BOD, and ∠BOC = ∠AOD.
Straight Line Property
If the exterior arms of two adjacent angles lie on a straight line, their sum is 180°.
Example:
∠AOC + ∠BOC = 180°,
The exterior arms OA and OB are in the same straight line, i.e., AOB is a straight line.
Example 1
Two straight lines AB and CD intersect at point P. If angle BPD = 54°, find, giving reason:
(i) When two straight lines intersect each other, the adjacent angles are supplementary.
=> ∠APD + ∠BPD = 180° => ∠APD + 54 ° = 180°
=> ∠ APD = 180° − 54°
= 126°
(ii) When two straight lines intersect each other, the vertically opposite angles are equal.
=> ∠APC = ∠BPD = 54°
Example 2
The figure given alongside shows two adjacent angles AOB and AOC whose exterior arms OB and OC are along the same straight line. Find the value of x. 
Solution:
Since the exterior arms of the adjacent angles are in a straight line, the adjacent
angles are supplementary.
∴ ∠AOC + ∠AOB = 180° => 2x + 10° + 70° = 180°
=> 2x = 180° - 80°
=> x = `"100°"/2`
= 50°
Key Points Summary
-
Adjacent angles: Side by side on a straight line, sum to 180°
-
Vertically opposite angles: Across the intersection, always equal
-
Every intersection of two straight lines: Produces four angles—two pairs equal, two pairs adjacent
