These notes for CBSE Class 9 Mathematics, Chapter 6, "Lines and Angles," provide a comprehensive overview of fundamental geometric concepts. The chapter begins by defining basic terms like point, line, line segment, and ray. It then delves into various types of angles, including acute, right, obtuse, straight, reflex, complementary, supplementary, adjacent, and linear pairs. The notes also explain vertically opposite angles formed by intersecting lines. A significant portion is dedicated to transversals intersecting parallel lines, detailing properties of corresponding angles, alternate interior angles, and interior angles on the same side. It also covers conditions for lines to be parallel. Key theorems on triangles, such as the sum of angles being 180 degrees and the exterior angle theorem, are included. These notes are ideal for students preparing for their exams, offering clear explanations and essential concepts for revision.
Last optimized 10 Aug 2026
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(1) Point - We often represent a point by a fine dot made with a fine sharpened pencil on a piece of paper.
(2) Line - A line is completely known if we are given any two distinct points. Line AB is represented by as . A line or a straight line extends indefinitely in both the directions
(3) Line segment - A part (or portion) of a line with two end points is called a line segment.
(4) Ray - A part of line with one end point is called a ray.
(5) Collinear points - If three or more points lie on the same line, they are called
collinear points otherwise they are called non-collinear points.
Types of Angles -
(1)Acute angle - An acute angle measure between 0o and 90o
(2) Right angle - A right angle is exactly equal to 90o.
(3)Obtuse angle - An angle greater than 90o but less than 180o
(4 )Straight angle - A straight angle is equal to 180o.
(5)Reflex angle - An angle which is greater than 180o but less than 360o is called a reflex angle.
(6)Complementary angles - Two angles whose sum is 90o are called complementary angles.
(7)Supplementary angle - Two angles whose sum iis 180 are called supplementary angles.
(8)Adjacent angles - Two angles are adjacent, if they have a common vertex, a common arm and their non common arms are on different sides of common arm.
(9)Linear pair - Two angles form a linear pair, if their non-common arms form a line.
(10)Vertically opposite angles - Vertically opposite angles are formed when two lines intersect each other at a point.
TRANSVERSAL
(a) Corresponding angles
(b) Alternate interior angles
(c) Alternate exterior angles
(d) Interior angles on the same side of the transversal.
•If a transversal intersects two parallel lines, then
(i) each pair of corresponding angles is equal.
(ii) each pair of alternate interior angles is equal.
(iii) each pair of interior angle on the same side of the transversal is supplementary.
• If a transversal interacts two lines such that, either
(i) any one pair of corresponding angles is equal, or
(ii) any one pair of alternate interior angles is equal or
(iii) any one pair of interior angles on the same side of he transversal is supplementary then the lines are parallel.
• Lines which are parallel to a given line are parallel to each other.
• The sum of the three angles of a triangle is 180o.
• If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
Q.1 In the given figure, The value of y is
(a)10o
(b)40o
(c) 36o
(d) 45o
Q.2 An exterior angle of a triangle is 750 and its two interior opposite angles are equal. Each of these equal angles is
(a) 105o
(b) 50.5o
(c) 52o
(d) 37.5o
Q.3 The compliment of an angle ’m’ is:
(a) m
(b) 90o+m
(c) 90o-m
(d)m ×90o
Q.4If one angle of a triangle is equal to the sum of the other two equal angles, then the triangle is
(a) an isosceles triangle
(b)an obtuse triangle
(c)an equilateral triangle
(d) a right triangle
Q.5 In the given figure form a linear pair if a-b = 100o then a and b are
(a)120o, 20o
(b) 40o, 140o
(c) 50o, 150o
(d) 140o, 40o
Q.6 Angle of a triangle are in the ratio 2 : 4 : 3. The smallest angle of the triangle is
(a) 60o
(b) 40o
(c) 80o
(d) 20o
Section - B
Q.7 Two adjacent angles are equal. Is it necessary that each of these angles will be a right angle? Justify your answer.
Q.8 In the following figures which of the two lines are parallel and why?
(i) (ii)
Q.10 In the fig. Q.11 Sum of two angles of a triangle is 90o and their difference is 50o. Find all the angles of the triangle.
Q.12 In the adjoining figure, .
Section - C
Q.13 In the given figure AB and CD intersect each other at O. If find the value of x,y and z.
Q.14 Prove that vertically opposite angle are equal.
Q.15 In the given figure
Q.16 In the given figure are bisectors respectively find
Q.17 The angles of a triangle are in the ratio 2: 3: 5 find the angles of the triangle.
Q.18 Find x and y in the following figure.
Q.19 In figure find x.
Section - D
Q.20 Prove that sum of the angles of triangle is 180o.
Q.21 Prove that sum of the angles of a hexagon is 720o.
Q.22 The angles of a triangle are find the value of x.
Q.23 In the given figure, AD and CE are the angle bisectors of respectively If Q.24 A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angle so formed are parallel.
Answer :
(1) b
(2) d
(3) c
(4) a,d
( 5) d
(6) b
(9) 65o
(10) 115o
(11) 20o, 70o, 90o
(12) 95o
(13) 84o, 21o, 48o
(16)90o
(17) 36o, 54o, 90o
(18) 48o, 12o
(19) 30o
(22) 100o
(23) 135o
The notes define fundamental geometric terms like point, line, line segment, ray, and collinear points.
The notes cover acute, right, obtuse, straight, reflex, complementary, supplementary, adjacent, and linear pair angles.
A transversal is a line that intersects two or more other lines. The notes explain its properties when intersecting parallel lines.
When a transversal intersects parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side are supplementary.
The sum of the three interior angles of a triangle is always 180 degrees.
The exterior angle formed by producing a side of a triangle is equal to the sum of the two interior opposite angles.
Yes, these notes cover the key concepts of Chapter 6, Lines and Angles, for CBSE Class 9 Maths, making them ideal for exam revision.
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