CBSE Notes For Class 9 Maths Chapter 6 Lines And Angles

1.0Introduction to Lines And Angles

The two basics in geometry are lines and angles. Whenever two lines meet, angles are formed, and relationships from which we can describe shapes and their properties are provided. By studying the relationship of lines and angles and how these interact with each other, we can solve geometric problems and prove important theorems. 

2.0CBSE Class 9 Maths Chapter-6 Lines and Angles - Revision Notes

What properties do these angles and lines have? 

Axiom-1: If a ray stands on any straight line, then the sum of two adjacent angles so formed is 180°. This property is more commonly known as a Linear pair of angles. 

Example: In the given figure, AB is a line that is cut by line CD. ∠ 1 = 75°, find ∠ 2. 

Solution: ∠ 1 + ∠ 2 = 180 (Linear pair of angles) 

75° + ∠ 2 = 180° 

∠ 2 = 180° - 75° 

∠ 2 = 105°.

Axiom-2: If the sum of two adjacent angles is 180° degrees, then the non-common of the angles forms a line. Commonly known as the reverse of a linear pair of angles. 

Example: In the given figure, ∠ DOA + ∠DOB = ∠ AOC + ∠BOC. prove that AOB is a line.  

Solution: ∠AOC + ∠BOC + ∠DOA + ∠DOB = 360 (sum of all angles) 

As it is given that ∠ DOA + ∠DOB = ∠ AOC + ∠ BOC

Hence, 

∠AOC + ∠BOC + ∠AOC + ∠BOC = 360 

2∠AOC + 2∠BOC = 360

2(∠AOC + ∠BOC) = 360 

∠AOC + ∠BOC = 180 

Hence, AOB is a line by reverse of a linear pair of angles. 

Theorem 1: Vertically opposite angle theorem: If two lines intersect each other, then the vertically opposite angles are equal. 

To prove: ∠ 1 = ∠ 3 and ∠ 2 = ∠ 4. 

Given: l and m Both are lines intersecting each other. 

Proof: l and m are lines, hence by linear pair axiom

∠ 1 + ∠ 4 = 180 

∠ 1 + ∠ 2 = 180 

∠ 1 + ∠ 4 = ∠ 1 + ∠ 2 

∠ 4 = ∠ 2 

Similarly, ∠ 1 = ∠ 3. 

Theorem 2: In maths, Lines parallel to the same line are parallel to another line as well. 

To prove: line a is parallel to line c. 

Given: line a is parallel to line b and line b is parallel to line c 

Proof: Line A is parallel to line b hence, 

∠ 1 = ∠ 2 (corresponding angle of parallel line) 

Line b is parallel to line c hence,

∠ 2 = ∠ 3  (corresponding angle of parallel line) 

Hence, 

∠ 1 = ∠ 3 

So, line a is parallel to line c by the reverse of the corresponding angle of the parallel line. 

3.0Solved Problems 

Question 1: In Figure, AB, CD, and EF are three lines concurrent at O. Find the value of y.

Solution:

∠ EOA = ∠ BOF = 5y

∠ EOC + ∠EOA + ∠AOD = 180 (Linear pair of lines) 

2y + 5y + 2y = 180 

9y = 180 

y = 20 

Question 2: In Figure, x = y and a = b. Prove that l || n.

Given: x = y and a = b 

Solution:

x = y

Hence, l || m (converse of corresponding angle axiom) 

Given that a = b 

Hence, m || n (converse of corresponding angle axiom) 

l || n as Lines parallel to the same line are parallel to another line as well. 

4.0Key Features of CBSE Maths Notes for Class 9 Chapter 6

• The notes are aligned with the latest CBSE curriculum. 
• The notes consist of solved problems and theorems to get a better understanding of lines and angles. 
• Visual aid is provided to make every concept easy to understand. 
• These notes are ideal for self-learning as they are explained clearly and easily.